The type cover issue may be a failure of the contact points. You can try cleaning both the cover contacts and the tablet cover contacts. The bad news from my limited experience with covers is that the "hinging" / flexing of the cover especially backwards tends to break the tracks in the cover right at the contacts and is not repairable. IF you are handy with a multimeter, you can check the contacts doing a resistance measurement on the "pairs". I don't recall what they are but you can find them in a search on the net. Anyway, the "pairs" should all have some form of reading / resistance and not be totally open resistance (infinity). IF they measure ok, you can do a similar check back towards your tablet (with it powered off). Again, you should see some form of resistance and not an open (infinity).
Of course, if you have friend, with an equivalent / compatible device, you could try swapping to test things.