try un-installing the drivers then reboot and then connect the KB and let your SP3 re-install it and it should work afterwards
My type cover stopped working last month too (after an update), and none of the above helped (except I hadn't done a full reset yet, was planning to do so this weekend) and I just installed System Firmware Update - 1/19/2016 and now it's working again!
All I know is for over a month my type cover, auto rotate pen didn't work. I mostly have been using my SP3 connected to a monitor/keyboard/mouse but as a tablet it was pretty much crippled. I was going to do a rest this weekend. The pen turned out to be a stuck side button which I popped out with my fingernail.Hey, I just got this update too...any idea what it is all about?