I've also long held the suspicion that Kvothe no longer has the ability to use sympathy, as he either locked it away in his own mind (much like how he hid the stone in his mind from himself in early parts of the book) or he did something regarding changing his name so he couldn't use it (on purpose or on accident).
As for what's in the chest, I think it has something to do with his inability to use sympathy. It also likewise requires some piece of his lost self to open, well beyond the keys, that he oh so cleverly locked inside. How to open a chest if the key is locked within... Classic Kvothe puzzle.
It seems obvious from the start that the character Kote is not Kvothe. While our third person narration reveals Kote to be a more than capable character (ie. he deals with the Screalings), the man is certainly not the "Kingkiller" by which we've be led to identify him as according to the Chronicler--I think this is probably another detail we'll find Rothfuss toying with before the series end. As I've already mentioned, the man does a great job of messing around with narrative reliability. Whether or not Kvothe is a reliable narrator would be a thread post in and of itself.
With that said, after two novels I'd love to put forth the outrageous theory that Kvothe is in the chest. I have a loopy Denna theory that ties all of this together, but with that aside I'll try to simplify it to the notion that at a certain point between the orphaned prodigy to the unrivaled sorcerer, Kvothe changed; by which I mean that the "name" by which one might "call" Kvothe was altered by such an event as to alter the "essence" upon which might have drawn were they to "call" on "Kvothe" (ie. the same way Kvothe manages to call on the wind, intermittently, and even then only certain aspects of the wind). As such, I think Kvothe, having realized there was an alteration in his character, performed some type of magicry to divide himself from himself.
It is very possible I am wrong,so I'm hoping Rothfuss is cleverer than I already attribute him to be and has something surprising and unexpected planned for the entire interlude storyline he already has going. Bast isn't hanging around for fun I'll say that much