I had pretty much same issue with a 13” MacBook Pro mid 2012.
Solved it by pulling the keyboard ribbon cable out of the ZIF socket.
Refer to step 11 of this guide:
[guide|10377]
I noticed some of the connection pins in the ZIF socket where gummed up with what looked like dust. I carefully cleaned the pins with a cotton bud dipped in alcohol. Also cleaned the connections on the keyboard ribbon.
Reinserted the keyboard ribbon. Turned machine back on and everything is now working normally.