The AK can drive the HE400. I have that myself. I like the pairing in general, but at times it comes close to maxing out the volume of the AK100 - if you have something that is really low in volume.
The AK is good with 320 files and will sound better than your iphone setup amped in terms of precision and clarity. You may get more impact though with your current setup. The unamped stock AK100 has a clean precise, smooth sound signature. I don't think this would necessarily be the best for the type of music you listen too. It will deliver your bass in a very clean, tight and precise way though. If you like that, then this would be good for you. If you want the bass to be more impactful, then you would probably consider something else.
If you're having to push your volume near full for decent listening levels you're not getting the best out of your headphones, get a nice little amp to help drive those puppies. Don't have to spend too much money for decent power these days either.