I have finally fixed it.
I got all of the loose wires re-soldered. Everything is functional.
I also bought x3 3800mah batteries and replaced the old ones! I checked voltages and all before soldering them to the console's wires. The battery life used to be 1 hour, now it's 3-4 hours. I closed up the console, then hot glued the edges at the bottom of it's shell to avoid hands squeezing the shell together while it's being used. No over-heating going on whatsoever.
Best part for last: I hot glued a silicone pad onto the joystick.
I have been playing it here and there for about two weeks now, and I think it is safe to say that this thing now works like a champ.
Thanks for all of the help and advice! Couldn't have done it without this forum. I learned a lot from this.