Your engine uses a Kohler 20 098 11-S starter. There is no internal solenoid. If you follow the large cable back you will find an external solenoid. Probably find it under the seat if you remove the battery. In this case, from your description, my experience tells me you only need a new bendix starter kit. This kit includes a bendix gear, thrust or bendix washer, spring and retainer. Any good small engine repair shop should be able to get this for you.