Rebuilt cheap starter motors sometimes don't last very long. The brushes could be stuck. Try taping on it with a hammer.

There is a blue wire that connects to the starter solenoid. Remove it and place a test light between it and a good ground. Have a helper move the ignition switch to the starting position. The test light should illuminate or the meter should show 12v. If it does not then you have a problem in the neutral safety switch or the ignition switch circuit. If the blue wire is not supplying 12v chances are when you carefully run a jumper wire from the positive post of the battery to the terminal that normally has the blue wire on it the starter will engage (if you do this do it with the key in the off position so the car will not start with you down in the engine bay- could be a day spoiler....)

If the test light shows 12v when the switch is moved to the starting position but the starter is not engaging the solenoid might be defective but not likely. Check that the battery posts and cable ends are clean. Clean the positive cable at its connection on the starter. Clean the negative cable at its grounded end on the chassis. Make sure that both cables are not showing corrosion up into the insulation at either end.

Randy