When you say thermostat wire, you mean the copper wire running to the dial thermostat? That could easily be repaired. If you mean the metal tube that runs to the coil (part that freezes) then that is another story entirely. The fridge won't turn on with the thermostat wire broken because the thermostat controls when the compressor comes on. Join those wires back together and it will work again. If it broke off of the solder connection then it can be resoldered. Honestly the thermostat could be bypassed altogether if you are going to have an external temperature controller and don't ever want to use it as a functional mini fridge again.