blue - that coil on the switch is supposed to get hot when the switch is set to less then full speed - that's it's job, to reduce the juice to the fan motor - not getting hot may be due to it being burnt out, or it may be due to the motor no longer drawing any current
you can run a jumper temporarily from the hot side [input] of the switch to the motor lead, if the motor runs the problem is the switch, if it doesn't, the problem is the motor
Bill