In the naval shipyard, we used to check 3 phase motors for loose/broken rotor bars using a "simulated single phase growler test." It's actually fairly simple to perform.
The stator is powered with a single phase, and vibration at 50/60 Hz (or maybe twice that) is monitored with an accelerometer as the rotor is slowly rotated. As I recall, the vibration level will drop as a broken rotor bar passes by the energized stator section. Make several rotations, and if a drop in level is noted at the same point in several rotations, there's a loose/broken rotor bar.
Using this technique, identifying which rotor bar is the problem would be tough, but it's a way to identify a problem.
I think your problem is with a single phase motor, but this technique may be a useful tool for some.