So the power input (the plugin where the AC adaptor cord inserts) is broken on my SNES. Specifically, the plastic "tube" that holds the cord in place broke, leaving the two exposed contacts dangling. 

It's still possible to turn on the SNES by rotating and applying force to the cord from a certain direction. I've jury-rigged the thing to work by clamping the AC cable to the video cable, but I'd prefer something more permanent.

I have a second SNES with an intact power input, but the unit has some sort of weird graphical problem that I can't figure out (and yes, I have thoroughly cleaned it). I disassembled both units to see if I could switch the intact input to the other SNES, but that doesn't seem possible without de-soldering some contacts, and that's beyond my capability.

Any suggestions?

Yup I fix them all the time.
desoldering and replacing the back is the solution.

If you can't do it, several of us can

Replacing that whole back piece is the only option afaik. Maybe you can find somebody local to help you with the swap.

I pulled out the metal tabs and drilled a hole then soldered in a Molex plug