Never heard of a post '08 KLR burning oil, but a 685 kit would solve that problem as well as smooth out the vibrations during high (for a KLR) RPM cruising.
Burnaby Kawasaki (604 525 9393) would be a good source of info
I was in a local shop this week and they were working on a newer KLR that smoked so bad it would kill every bug around it in a 30' radius. Apparently Kawasaki has been replacing the piston/rings/etc under warranty. They were going to on this bike anyway. It wasn't the Kawasaki shop I was at, but it was going there soon.
Ther dealer stated that unless I burned more than 1l per 1000km there would be no warranty service. As a career licensed mechanic I am very weary of warranty repairs as they are done on the cheap for the most part.