I have the latest kali dist 1.0.6 installed and running from SD Card on the Samsung Chromebook.
Basic wireless works fine now.

I have read that wireless injection does not work, and I verified that the driver will not go into monitor mode.

Question -

Anyone have a modified driver that does work with the obscure on-board chipset?

If not, what small wireless USB (w/built-in antenna) do you have running that does work with injection? I have seen some of the recommendations, but I was looking for someone with hands-on experience with something they know works.