I have had this same issue for a while. I finally spent my whole day today fighting with it and finally got my Surface Pro 4 to work with the type cover again. I don't know if this will work for any of the other surface models. Step one I uninstalled all
of the windows updates that I could by going to settings>update and security>Update History>Uninstall Updates. I uninstalled any windows 10 update it would let me. I then restarted and allowed it to begin auto downloading the updates again. During that time
I went into my device manager by right clicking the start menu and selecting device manager. Under firmware I uninstalled "surface embedded controller firmware", "surface system aggregator", and "surface uefi". I then installed all the updates that windows
10 needed and restarted however many times I had to. When that was done I went back to the device manager and went to those same 3 firmwares and right clicked them and told them to update. Each one downloaded a new update. When all three were done a windows
alert popped up telling me that the type cover driver was being installed and set up. I then had to disconnect and reconnect the type cover before it finally recognized it and turned on.
I completely lost it when it finally worked because I had kind of accepted that I paid for an overpriced tablet. Hope this works for you. You may need to uninstall and update the firmware more than once because I did, but it was because I was doing things
in a kind of random order hoping something would work and when it finally worked it was when i uninstalled and reinstalled the firmware after the win 10 updates.
Good luck and I hope this works for some of you. Why microsoft can't seem to get this to automatically occur without so much user input is beyond me. Makes me yern for the days of manually installing and updating drivers.