If the sprocket disengages from the flywheel, then there is no problem with the solenoid. I bet its the switch. My starter would engage once you turn the key from the off position and would continue to engage even when the engine had started. It was like i never let go of the key. I still had to replace the starter because the bendix wore out. The starter ran while i was driving.

Remove the two wires(violet and red) on the RH wheel well(C105). Secure the terminals with electrical tape. Turn the ignition key to turn on the glow relay and start the car by bridging the two terminals in C105(I forgot which two). If it starts normally you just confirmed its an ignition switch problem. I dont think there is any relay in the starter.

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'85 300D Cal 280,000 miles
'14 GLK 350 60000 miles