Well, they were out of JB Weld at the store. So I bought something called Faststeel, which is a "steel-reinforced epoxy putty." It was only $3 so it's not like it was a waste of money really. But it didn't do the trick. It held great for a day, but at the end of the day yesterday it broke. I'm not sure whether it didn't bond properly to the various materials around the hinge, or whether it failed because of heat stress, or what. But we're back to square one. I do have to go out today in the direction of a different home improvement store, so I'll check again for them.
But Smittygirl had a thought. What about getting some little brass hinges and bonding them to the plastic frame? Then the lid would still be able to close, and I could come up with something else to protect the wiring. Might work. Any thoughts? Will JB Weld bond brass hinges to plastic? Hot glue perchance (ha ha! Battleroach 2000 lives!)?