The K2 and the DX use the same CPU as the K3 (arm1136jf-s / vfp), with a slightly older Kernel (~2.6.22) (and half the RAM, if I remember correctly?).
EDIT: AFAIR, the cpuinfo is indentical (DX/DXG/K2/K3), but I may be wrong

.
Keep in mind that I had some weird surprises when building lua/kpdf with a hardfloat TC (namely, a completely messed up lua interpreter). I never investigated much further than that, so it may be a stupid mistake on my part

.
EDIT: On the other hand, it works flawlessly with my latest softfp TC, so, there's that

.
(Check the
kindle-x-compile-tc.tar.gz tarball for more details on my K2/K3 TC).