One of your CIA chips may be faulty.
They are notoriously fragile. I think they may be some of the few chips on an A1200 that aren't surface mounted precisely for this reason.
Someone will be along to confirm or (probably!) criticise what I say.
It may be possible therefore to swap out CIAs from a known working A1200 and see if this sorts the problem.
As an aside it is essential to not plug in or remove any peripheral (even a mouse or joystick) whilst the machine is on. That way you will get the best life out these chips.
Edit to add: I've just checked some photos of various A1200 motherboards and it looks like the CIA chips are NOT socketed to it will be a tougher repair if this is indeed the fault. :-(
