Maginnis
I was taking a TT keyboard apart just the other day, which is basically the same keyboard as the Mega STe, so thought I'd throw in my two pennys worth. The keyboard is a fairly simple affair consisting of a plastic membrane like circuit, with each key acting like a switch. I would certainly open it up as you can't do much damage. There's a connector that resides under the plastic case containing the keys and this circuit, check that this is still in situ. If you still get no joy take apart the section containing the keys. You have to push back a load of tabs on the underside to do this, I found it best to start at one end and work to the other loosening tabs as you go. The plastic circuit in mine had some gunk in, which had somehow coated the plastic and degraded the circuit. It didn't look a particularly easy thing to fix in mine and I would imagine wouldn't be opening to soldering as it would probably just melt. The joystick ports are soldered into a flimsy bit of proper circuit board. They seem to suffer from the same problems as a standard ST keyboard, as in they come loose, and you would have to desolder them and then check the connections. I can't see what else might go wrong other than some corrosion on the circuit somewhere or a blown chip.