I don't really have a good solution but felt bad that no one else had one either. Toshibas seem to have a lot of electrical connection issues, ranging from power cords that barely miss connecting to their sockets to your battery issue to overheating. They also appear to be harder than average to dis-assemble for repair. Check out these instructions if you have time:
Toshiba Satellite M35X. How to fix most common laptop problems. - Laptop Disassembly Help.
I think if anyone is good at repairing Toshibas, they probably are laying low because they are so busy. We solved our Toshiba problem by buying a very cheap and tough Gateway.