It's not clear from your pictures how the motor is connected to the circuit board. You want to determine if the motor is fried, or if the the driver on the board is fried. If you can determine the driving voltage (probably 120v, but I hate to ask a newbie to play with that), then you can try driving the motor directly to make sure it works. Otherwise, you could try to replace the component driving it. It's admirable that you'd like to learn and reduce waste, but I agree with the above -- chances are that even if you can determine that it needs, say, a new high-power driver, you could spend a long time trying to find a cheap replacement. Or a "lot" of money ($10 being a lot in this case, if used ones are available for that price).
If you want more help, take it more apart and provide lots and lots of clear pics. No promises, as I am not a Devry certified electric appliance repair man.