I don't know if DE will kill them, but it does prevent them. Once you get rid of the lice, treat your goats daily with DE. You can put in there food, bedding, in dishes, where ever you like. I use it, along with my mom and NONE of our animals have every had lice, fleas, mites, worms, etc. etc. I do have oral wormer on hand, but to this date have NEVER used any of it.
If you can contain the goat to one area until the lice is gone treat the area the goat was contained in too.
I just seen on the DIY channel one of there DIY FYI tips. And for lice/fleas it says to vaccum up area, put moth balls in vaccum bag. Also to sprinkle salt on the ground. The fleas/lice will die from the salt. Could try something like that.
Best of luck.
Shawna