biggsteve
Gone But Not Forgotten
From what you have described it appears to be a 3 speed AC motor.
It sounds like 2 of the speeds were shot when you say it would only run on the 3rd speed. It sounds like the motor completely went out when it burned the cord in two.
Before you plug it in again I would suggest (with the fan unplugged) taking a volt/ohm meter set on ohms and checking the resistance between the 2 wires of cord. With the switch set on, and still unplugged, you should read relatively low resistance. 10 ohms or less would be my guess. If you get no reading the motor windings are shot. If you get a good reading, check each wire to the metal housing of the fan. You should get no reading. If you get a low resistance reading, the motor is shorted to ground (the housing).
Please do this, or have someone do it, before trying to run it again.
much too complicated for a 'lay' person to do.
she should take it to an appliance repair shop to be checked out.
then go from there. imho.