After SpeedyG's excellent advice about my keyboard troubles, my ][+ can now be semi-used.
However, it's revealed a new problem : it can't read disks!
I have tried 2 working Disk II controllers in this II, with known working drives.
It powers them up and makes the usual clacking sound, but does not make any disk reading sounds.
These drives work perfectly in my other machine.
I tried the controller card in various slots, it doesn't work in any of them.