If the picture you have attached is similar to the starter on you car, then the power to the starter motor is only delivered if the solenoid operates, so the solenoid must be working.
However, the shift lever labelled in the picture, that pushes the pinion forward to engage with the flywheel may be broken, that would explain your problem.
Your problem could also be explained by the pinion being siezed on the shaft it slides on, where all the motion of the shift lever is being absorbed by the spring behind the pinion.
It is also possible that the teeth have been stripped off the pinion or the flywheel. If the flywheel teeth are damaged, you could try nudging the car forward a little with it in gear so that the flywheel moves different teeth next to the starter pinion.
My hunch would be that the starter pinion has siezed on the shaft, so that it is not being pushed toward the flywheel. You may be able to tap the rear end of the main starter motor shaft with a hammer to free it off, otherwise you will need to remove the starter and lubricate the shaft.