So maybe you got a good deal on the pump, but just bad luck on the motor controller. All is not lost if you can verify that the pump motor still works.
Maybe you can purchase a PWM controller from one of the many Asian sources available on eBay. What to get would depend on what kind of motor the air pump uses. That will require some experimentation on your part. I doubt very much that the motor uses anywhere near the 110/220 VAC, 50/60 Hz specified as the input voltage. It is probably a permanent-magnet, brushed, DC motor that would run fine on anywhere from 2 V DC to maybe as much as 24 V DC.
So, I would open it up, find the motor wires, disconnect them from the faulty speed control, and start connecting the motor wires to dry-cells, connected series, until I found a voltage that ran the motor at whatever maximum pumping speed you wanted or needed. Pay attention to polarity and make sure the motor rotates in a direction that results in pumping. Then order the appropriate PWM controller after measuring the motor voltage and motor current at the maximum pumping speed.
If you have any electronics skills (this IS a hobbyist forum) you can make your own PWM controller... probably not as cheap as you can buy one from Asia, but a lot more fun, and a learning experience to boot. I would suggest using a step-down transformer to get into the right ballpark for the DC motor supply, just for safety reasons. It is dangerous to mess around with mains power applied to low-voltage circuits without a transformer to isolate and limit the current available to the maximum necessary for circuit operation.
If motor insulation is adequate, you can theoretically use PWM at the line voltage (after rectification) to control motor speed, selecting a maximum duty-cycle for the PWM wave form that does not overpower the motor. This is a risky and dangerous approach in my opinion. You can easily "let the smoke out" and ruin the motor by accidentally operating with too large a duty cycle. Also, it places a high voltage stress on the motor windings that may cause early failure if you are lucky, or immediate failure if you are not lucky. So, use a step-down transformer, rectify the secondary voltage to the proper DC voltage level, then use PWM with a 0 to 100% variable duty cycle to drive the motor.
Edit: Of course all the above depends on (1) your pump uses a brushed DC motor and (2) the motor is still operational and only the controller is bad. If you do the test Edd suggested, that will determine whether the controller is bad or not. Let us know what you find out and we can proceed from there...