What is the best way to improve armor class as an eldrich knight without having a high arcane spell failure chance, or taking up feats and the swift action? Is there such a thing?
If not what would you say is the best middle ground?
The optimization is based on your original spellcasting class. Some of them have the ability to cast spells in light armor with no risk of arcane spell failure. Assuming you did not do that, you need to find a way to replace the armor and shield bonuses to AC.
For the shield bonus, go for mithral to reduce arcane spell failure chance to nothing.
For the armor bonus, you can choose between a haramaki or silken ceremonial armor (both with arcane spell failure of 0%) or bracers of armor. Check prices and availability to get the best bonus. If speed of movement is not important to you, throw an armored kilt on top of that armor for an additional +1 to AC. Alternatively, if you are a high enough level caster, you might get by with castings of Mage Armor.
Then focus your resources on the other contributors to AC -- amulet of natural armor, ring of protection, belt that improves dexterity, and so forth.
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IMO, Arcane Armor Training isn't that bad - once you get up to Mastery at 7th level, you can wear Mithril Breastplate with 0% spell failure and Mithril Full Plate with only 5% spell failure chance. You can always use Still Spell to remove spell failure chance, as well, if you want to cast quickened spells. And it almost never overlaps with Arcane Strike, since you'll rarely be casting and attacking on the same turn.
If you want to use a shield, a Mithril Buckler is probably the way to go. Note that if you use the hand with the buckler to cast, you lose the buckler bonus until next turn, so... watch out for that. Nonetheless, enchanting a shield is going to be a relatively cheap way of getting AC at higher levels.
Otherwise, Mage Armor or Bracers of Armor (at high levels) are your best bets. Or just make your base classes Empyreal sorcerer and Sohei monk, and you get your casting stat to AC!
if you use magus or bard levels to build into it, you can cast in some armors without ASF, or the tier 3 archmage mythic ability "component freedom" (choosing to ignore somatic components, which in turn doesnt roll for ASF) can help nip the problem in the bud.
Still spell. It's a fine feat.
You have enough options to eat up your swift action.
What is the best way to improve armor class as an eldrich knight without having a high arcane spell failure chance, or taking up feats and the swift action? Is there such a thing?
If not what would you say is the best middle ground?
See my thread started two days ago (Sun 10/13/2013) for several conversation points related to Eldritch Knight.
If you allow mythic progression: one tier 3 archmage ability has you covered --> Component Freedom. Unfortunately, you would have to potentially suffer through several levels with arcane spell failure, no armor, or only casting spells with no somatic compounds.
Outside of that, you have mithral armor for -10%, celestial plate armor -20% (not core but ask your DM if its allowable), and two feats for -10% or -20% (arcane armor training, arcane armor mastery). Mythic arcane armor training also eliminates the swift action activation (again not an option in every campaign).
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Magus Eldritch Knight with Dex focus and agile weapons. You get the ability to cast in light armor, so you can cast in a mithral breastplate with the armor expert trait with no penalty. If you exceed the max dex of that, save for celestial armor and you get up to +8 Max dex, with a +9 base, for a total of +17 BEFORE you add more enchantments, rings, amulets, shields, etc. Works the same with bard.
Or you could wear darkleaf lamellar leather with nice max dex and a 5% ASF, which is laughable, or (as Reynard the Fox said) have a level of monk and be Empyreal for your casting stat to AC.
Or, you know, the built-in solution to everything in the game, magic.
I play an EK and while I do wear armor (Mithral Chain Mail) and use Arcane Armor Mastery, the best defense are spells. Shield boosts my AC to a respectable 28 (at level 9) without interfering with a two-handed weapon. Other defensive spells help as well (Mirror Image, Displacement, Greate Invisibility, Stoneskin, etc.).
Also, if you are a Wizard, take an Amulet or Ring as bonded object so you can enchant it yourself for a cheap-ish AC boost.
Also, remember to cast long-duration buffs, like heroism, before putting your armor on. No spell failure chance.
Just take 8 levels of Hellknight Enforcer/Signifier and laugh as you cast in Mithril Chain shirt (or better yet Mithril Kikko Armor) without needing to use a swift action. If you DEX is high enough consider Bracers of Armor (which can be made with the best of all crafting feats, CWI) with a haramaki or silken ceremonial armor to hold any armor abilities you want.