Of course. I assume that your are fine taking the kindle apart to gain experience, I have one myself, I have not yet attempted disassembly though. One second thoughts, disassembley is not necessary, unless the battery requires replacement or you wish to integrate the charging system into the existing case.
You will need two inductively coupled cores. Either you can buy a prebuilt system, re-house it for greatest convenience, and potentially modify circuitry to specification or you can make your own. How you would make your own (efficient) induction pair, I am unsure, this would likely involve a lot of experimentation. Obviously, you would need some sort of pulse generation transmitter side, and rectification circuitry and smoothing received side. I found
this, all that I can say is it is excellent and this guy knows what he is talking about. The charging rate is not particularly fast however you may be able to increase the power through the transmitter transistor's collector - emitter.
What electronic components do you have, for example, on your desk. Pick them up and try and think of a novel use for them. It is hard to know what others are capable of building when their inventories are unknown. I know exactly what I could build, because I know what experience I have and the parts I have access to. For all I know, all of your parts could have been used up in earlier projects and you have only a few LEDs lying about or you could have an effectively unlimited supply of just about everything.
One project that I played with other the weekend with was ambient EM emmision harvesting. I am about to add the project to the project logs section. You need a ground connection, a foot square of aluminium foil, a full wave rectifier, a smoothing capacitor and a load. Connect one of bthe ac input sides of the rectifier sides to ground and one to the aerial. Connect the capacitor to the rectifier output terminals. The full wave rectifier requires four diodes, Google it.This does not produce a lot of power, but is an interesting proof of concept. The only reason that I made this circuit was because I had a roll of foil in my room and knew that I had a large supply of tantulum capacitors and diodes.
Any of this sound at all interesting?