You might have a bad type cover. Mine died a year ago. No matter what I tried my SP3 just wouldn't recognize it. I finally scheduled time at the MS store techs. We played around with my SP3 and the type cover for quite awhile, nothing worked. The tech was about to go grab a replacement SP3, thinking it was at fault... But I tried my type cover on a display Surface... didn't work. But the display type cover worked on my SP3. We went around to all the Surface's to just make sure, but yep... the cover was dead, dead, dead.
The sad part was that I found out that the extended warranty didn't include the Display Cover, so I had to pony up and buy a new one. I tried to get them to take the bad one, and send it somewhere to do a failure analysis on what broke, but they just told me that they don't do it anymore. They just toss/recycle now.
I still have the dead cover... one day I will dissect it myself for curiosity sake.