@dpavlin: Yifan's writeups on this are here
(building) and there
(patching the update/recovery).
Seaniko7 also did some stuff with Kindle kernels, here
a small patch for the 3.3 kernel to at least *build* on a recent system.
If you intend to flash *everything* (kernel + modules + support files), I think any (working
) toolchain will do, but if you want to mix & match, you'll have to use something mostly matching Amazon's TC (twobob might steer you in the right direction here, he's done it