If your a monk archetype that hasn't given up flurrying you could just be flurrying all the time with a temple sword. Much easier to deal with than using unarmed/natural with manufactured weapons.
I'm not sure how two weapon fighting works with unarmed strikes, but if your going to use two weapon fighting you do need the feats. However every monk but MoMS gets flurry so its not an issue for them.
Basically, a monk can make attacks with flurry-of-blows with any combination of weapons they have in any way they like. They do not need TWF to do this, as long as they are unarmed strikes or monk weapons.
So if your monk has a cold iron kama in one hand, a silver temple sword in the other, he can make his attacks in any order and with any weapon he likes. If he is 6th level, say, fighting an unknown creature, he could make his first attack with the temple sword. If that has no effect, he knows that silver is not this creatures vulnerability and can attack with the kama. If that has no effect, he could employ his unarmed strike and see if that has an effect.
You don't even have to declare this in advance, just make the attacks and decide which you will use on the fly.
This is one thing flurry-of-blows is good for.
I envision him being able to attack with his sword and throw a knee/elbow/punch, but am unclear on the game mechanics.
You'll have to ask your GM.
There's debate on whether you can two-weapon fight with unarmed strikes, though that might center around TWFing with only unarmed strikes (left fist/right fist) rather than one UAS and a manufactured weapon.
You should be able to use at least one unarmed strike with two-weapon fighting (it's specifically mentioned in the TWF rules), but how that will work with the monk special ability will require GM interpretation.
Personally, I would let it work. You can main-hand the sword, and use unarmed strike for the off-hand, and I assume the intent behind the monk text was to grant you full strength damage for the unarmed strikes (rather than forcing you to use the UAS as your main-hand).
Basically, just like a fighter with a sword and improved unarmed strike, but better.
Let me elaborate on my character. He is a duelist who uses an aldori dueling sword. I took a couple of levels of Master of Many Styles to add some more flavor to my character. I envision him being able to attack with his sword and throw a knee/elbow/punch, but am unclear on the game mechanics.
Master of Many Styles does not have flurry of blows, and you cannot flurry with an Aldori duelling sword anyway, so it's a purely academic question.
In the case of your character, he cannot flurry with his Aldori sword and gets no extra attacks, end of. If you have TWF, then you could, but that may impact some of your duellist abilities - using your unarmed strike may (in the opinion of your DM) count as "attacking with a weapon in your off-hand" - the weapon in question being unarmed strike.