Stock Kernel won't work with AOKP or any AOSP based ROMs for that matter. The stock ROM for AOKP is Faux kernel which isn't bad, but it also isn't my favorite. I would say that you should give Faux a try and see how it does, the battery life on this phone isn't as bad as others like my GNex which barely lasts a day with moderate use. If Faux (stock AOKP kernel) doesn't work out for you flash Lean Kernel, I had terrific battery life with it.
Lean Kernel is flashable through recovery and once you have it flashed, you can use Android Terminal Emulator to flash newer versions without opening recovery. Also, when downloading Lean Kernel, make sure that the newest AOKP B5 build is compatible with the newest CM merges over on Rootz. If you where on B4 and flahed Lean Kernel 1.5 your phone wouldn't boot because of that. I would start with Lean Kernel 1.4, I had amazing results with that.