have seen a few times behavior similar to what Daniel reports; the typecover is responding for a while but it is not stable - if you reboot it dies again.
It seems that I have successful resolved this, following these steps (if I remember correctly):
- first reset again the machine,
- then go to "Bluetooth and other devices", if the typecover is installed you can see "Surface Typecover" on you right hand-side. Most probably you will not able to see it, if this is so go to the next bullet.
- try "Add Bluetooth or other device",
I don't remember the exact steps after that, but for the last 3 days the Typecover is working fine. Just in case turn off automatic updates.
My experience trying to resolve this is that it is a driver (not firmware) problem
I hope my post helps