You picked Keen Edge for your only 3rd level spell known? What are your other Spells Known?
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I looks like you're building your character to be a Fighter anyway. If you want to boost your caster levels look at Prestigious Spellcaster. It cost 2 feats but you will get your caster lvl back.
If you want to bump up your ability as a fighter, and you are using power attack, take the spell Blade Tutor Spirit. It will be like a +2 to hit for you at this level. If you have the feats on your build you may also want to look at the Eldritch Scrapper archetype. Martial flexibility will be huge at higher levels and even more significant with Transformation (it ups your BAB for feats requirements). You immediately get access to Dedicated Adversary (+2 tohit, dam & some skills vs target). You can get your weapon enchanted with frost via Eldritch Heritage if that bloodline is important to you, but it won't stack with the icy burst on your weapon.
I would also consider learning the 1st lvl spell Vanish and the 2nd lvl spell mirror image if you are mixing it up in melee while enlarged.
You picked the right spells for the most part, so as not to compete with the crazy equipment you've been given. None of your class features really compare to a +10 weapon. Haste would have been another good choice, but keen edge is somewhat better since you can cast it long before a combat.
The level prior and first level of Eldritch Knight are the toughest. The first level is basically just a level of fighter, but it improves very nicely after you've paid that toll to get in. The low number of spells a sorcerer gets evens out in just a couple more levels to have enough variety to be truly effective. Make sure to take at least 1-2 metamagic feats so you can keep your options effective (Empower and Quicken eventually are top picks).
Basically i am finding myself playing a crappier Fighter, since my spells deal pretty much no significant damage to my enemies and my summons are getting destroyed with a backhand from every mob.
Summoning spells are not a very good choice for an Eldritch Knight. I would recommend retraining any of these you may have; they're pretty bad and a waste of your time in combat. Same goes for direct-damage spells (like Frigid Touch); if you're looking for that, go with the Magus.
Looking at your spell list, it seems you're a melee EK. Enlarge Person is a very good spell selection there, and I would build on that by using a reach weapon for maximum attacking reach. A size-enhanced EK with a reach weapon can be quite dangerous, and has a nice edge over the similar reach cleric build thanks to the Wizard/Sorcerer spell list (reliable access to flight is the big one!)
As for spells to pick up, I would say the Mirror Image spell would be great defensively. Create Pit or Web are nice for controlling enemy movements. As mentioned, Fly is a superb option for Eldritch Knights. With a 20 ft reach you are outside of high jumping distance of most opponents, meaning you're completely impervious to most normal-sized melee-only enemies. If your party doesn't have someone else who can cast it, Haste is exceptional.
As has been mentioned, the Prestigious Spellcaster feat is a must-have for EK's now. Buying back your lost spellcasting level is just too good to pass up, and will do more to help alleviate your spell level issue than anything else possibly could.
Exotic Weapon Proficiency in fauchard gives you a reach weapon with an 18-20 crit range (15-20 with keen edge.
Combat Reflexes + cats grace might be fun.
Eldritch knights are weakest in the early levels. Later on though you become a buffed up fighter that can actually do things outside of combat.
For feats have you considered the dimensional assault chain? Requires some investment but gives you pounce with style
Investing in a spellstoring weapon and shocking grasp could add some good damage. Also I recommend getting some quicken rods due to your equipment situation
For spells look for long duration buffs, swift action spells, and only 2-3 in combat spells
Lv1 Heightened Awareness, Lucky Number, Wave Shield, Darting Duplicate
Lv2 Resist Energy
Lv3 Spellsword, Ablative Barrier, Stunning Barrier Greater, Countless Eyes
Lv4 Dimension Door, Invisibility Greater
Lv5 Overland Flight, Possession
Lv6 Shadow Endurance
Lv7 Arcane Sight Greater, Shadow Body, Subjective Reality
Another build option is to build around the spell Telekinesis. Think I can dig up the link to an example build if your interested
The d20SRD doesn't list all the spells available on that table.
Lucky Number is a fun, useful little spell.
As everyone here has already said, playing an Eldritch Knight for the sake of being a Gish is kind of not a very good idea anymore, but it's not entirely a pointless idea. At a cost of just 2 levels of spell progression, an arcane caster can pick up 10 points of BAB in the course of their career, and three bonus feats. This is a very good boon for ray-slinging Evokers and Sorcerers and such, especially since those classes are a little starved for class features anyway.
If you don't mind throwing away your school powers or bloodline powers, and you do want some more BAB and some combat feats, Eldritch Knight is for you!
playing an Eldritch Knight for the sake of being a Gish is kind of not a very good idea anymore
This is the conventional wisdom, but I disagree with it. The Eldritch Knight is certainly hard to work with and very unforgiving for beginners, not to mention has an unfortunate valley of suck, but if you know what you're doing it's a functional and unique option that's still relevant today. And, most importantly, it really doesn't compete with the Magus for its niche. I actually don't feel the EK and Magus have all that much in common besides being arcane spellcasters that use weapon attacks.
The thing about the Magus vs EK comparison is that the things the Magus is good at are things the EK was never really good at to begin with. Direct damage spells like Shocking Grasp are a complete waste of action economy for an EK, who should easily be able to vastly out-damage such spells with his regular attacks. Meanwhile, the EK's biggest strengths actually lie in areas where the Magus is deficient: the broadest possible spell list and the ability to use a variety of weapon combat styles effectively.
I would love to see it unchained and actually get some class features (hopefully focusing on its versatility and adaptability), making it a more approachable and effective for a variety of players, but even in its current lackluster state it's a viable option for many concepts.
As everyone here has already said, playing an Eldritch Knight for the sake of being a Gish is kind of not a very good idea anymore, but it's not entirely a pointless idea. At a cost of just 2 levels of spell progression, an arcane caster can pick up 10 points of BAB in the course of their career, and three bonus feats. This is a very good boon for ray-slinging Evokers and Sorcerers and such, especially since those classes are a little starved for class features anyway.
If you don't mind throwing away your school powers or bloodline powers, and you do want some more BAB and some combat feats, Eldritch Knight is for you!
Agreed with Dasrak. Magus and EK both fulfill their own mechanical roles. There are archetypes/schools/bloodlines for Arcanist, Wizard, and sorcerer that are frontloaded and therefore benefit from going into EK. While it does give worse spell economy(until lv10) it offers the ability to gain higher level spells along with a slightly higher base attack bonus. EK can use all kinds of passive buff spells long before a fight begins in order to be, for the most part, a very strong fighter with much more utility. Meanwhile magus mostly offers Nova damage. EK, like most classes, does suffer from a period of relative weaknees. In addition it is one of the hardest to build properly. This gives it a bad rep sadly
I wouldn't play a Sorcerer Eldritch Knight. Sorcerers get great benefit from Class Abilities, and Eldritch Knight Sorcerers get none of those Sorcerer Class Abilities.
EK Sorcerer only works well if you're combining it with a martial class that uses charisma, such as the Paladin. Even then most people prefer the Dragon Disciple for this purpose, and if you've got the feats to burn to take Prestigious Spellcaster multiple times I'd definitely recommend that option of EK. Another build I've been looking at recently is using the Mysterious Stranger Gunslinger with Sorcerer into EK, which is second only to mystic theurge in terms of obnoxious valleys of suck but really matures into something awesome (cha-to-damage with near-full Sorcerer spellcasting - nice).
I had thought it would be interesting to play an Arcane Archer via levels in Eldritch Knight.
Most Arcane Archer builds will utilize the Eldritch Knight as part of their build, since you necessarily need a bit of filler (either at the start or end of your career, depending on when you qualify) that doesn't kill your attack bonus or caster level. EK fits the bill perfectly.
AA got a lot better recently with the Prestigious Spellcaster feat, which solves one of my biggest complaints with it. I'd honestly prefer the spellcasting of EK over its class features, but now if you have the feats for it you can have your cake and eat it too.
I've played a Ranger4/Sorcerer6/EldritchKnight7 from level 1 up. I hope my own experience can be helpful.
Started as Ranger because of max starting hitpoints, which made lower levels easier to survive.
Plus as a Ranger, I'm able to use CureLtWounds wands without making a UMD check, and I get an animal companion (usually a horse, or hunting dog, which lasts only a few days before getting splattered; replenish after the adventure).
My spells known were all a mix of buff, attack and utility.
The key to a Sorcerer is in your approach toward spells known. Since you have so few, you need to think about getting maximum return on investment, while considering a few facts
FACT 1 - if a low level spell is useful enough at minimum caster & charisma, it might not be worth picking. Get wands and/or scrolls.
FACT 2 - if the spell is a situational "just in case" spell (like "Remove Curse"), it's not a very good choice. Get a wand or scroll for it.
FACT 3 - You have many spells/day, and not many spells known.
To make the most use out of your spells/day, make sure your first spell of that level is one you will spam. That is, one you will use often. For example...
0: Acid splash
1: Magic missile
2: Invisibility
3: Haste
4: Dimension door
5: Telekinesis
6: Cold ice strike, or Disintegrate
7: Greater shadow conjuration
8: Greater shadow evocation
9: Timestop
Also, you want spells that are multipurpose.
SIDE NOTE: I just realized you listed "Cold Steel" as a spell, when in fact it's a bloodline power.
Therefore you are missing a level 1 spell known.
Also, Monster Summon I is (in my opinion) garbage.
Might I suggest from among the following 1st level spell choices:
EK Sorcerer only works well if you're combining it with a martial class that uses charisma, such as the Paladin. Even then most people prefer the Dragon Disciple for this purpose, and if you've got the feats to burn to take Prestigious Spellcaster multiple times I'd definitely recommend that option of EK. Another build I've been looking at recently is using the Mysterious Stranger Gunslinger with Sorcerer into EK, which is second only to mystic theurge in terms of obnoxious valleys of suck but really matures into something awesome (cha-to-damage with near-full Sorcerer spellcasting - nice).
I went Arcane->Sage for my EK so he's a smart fighter. I built my guy as a support buff/utility for his casting portion of it so its nice to have a plethora of spells to throw around for whichever situation we stumble ourselves into. My only offensive spells right now are shocking grasp, scorching ray, vamp touch (for spell storing weapon), and monstrous physique II. Everything else are buffs, defensive spells, or utility.