Ah. Can't just fix the wire eh? Most fans have a thermal fuse built into the motor windings so if it gets too hot that blows and there's no fix for that but that's not your problem.
Just grab a cheap fan even a little 6" clamp on one to move air around a bit. Have you got a small heater that can be run with just the fan blowing room air? Doesn't take much. That cardboard thing will help but is a real PITA.