According to the engine link (CV22S-75534) from your model # given, the new starter part # is PART NUMBER: 25-098-07-S (was 25-098-05-S). Google will be your friend. I have found that though more expensive- the oem Kohler starters seem better that the will fits, but ymmv. If you would like a second opinion on your diagnoses, feel free to run by the details

Better be sure that the problem is in the starter. Have it checked by a mechanic. And make sure you have the battery connected correctly. Red is positiv(plus), black is ground/ negative (-)! + and - signs are usually molded into the battery case somewwhere.
Many amateur mechanics have bought new lawn tractor starters, only to find out that the problem is not the starter, but something else. Also, new starters and other electrical equipment is not returnable, once you have hooked it up, tried it, and found the problem was not in the starter! UH-OH.
Most batteries as used in lawn tractors, etc. are connected as the minus sign goes to the frame ground, and the plus side goes to the starter solenoid.
Did you try jumper cables from the car or truck battery? Might save ya a Century Note! Also, i good cleaning of the minus cable, where it is connected to the tractor frame, is good to do. Check also for any disconnected safety switches, too.

It turned out we needed a new starter, battery and selonoid. Now the little tractor runs like a clock.

I guess after 12 years it was time for a tuneup.
Like most people a little help running goes a long ways.

Thank you all for you assistance.

Glad you have it running, but it's unlikely you needed all 3???

A relatively illiterate friend asked me to check his old Craftsman lawn tractor--said the starter wouldn't send the gear up the shaft to make the flywheel turn! Said he had bought a new battery, and when that didn't work, he then bought a new starter.
So, the first thing i did was try it, and he was right. So, i switched the connections, tried it and it started up and ran well! WOW!! I was King of The Hill, that day!

We had the satarter tested first at a rebulid shop and it was totally shot not able to rebuild. We had the battery tested and the cell was bad sot the battery could not be recharged and new opne was purchased. Tractor still did not start. This left the selinoid we tested this per insturctions given on this site and it was bad. Replace the selinoid and the tractor starts. So the little tractor needed a tune up after 12 years of service.
Thanks for all the help.

Yah, I have a Ranger pick up just like that. Do you store it inside? Probably best you got it all out of the way. A few minutes of trouble shooting may have found the cruddy connections on the solenoid. But after 12 yrs , if it didn't need it now, it could at any time.

You see that happen often enough. A starter gets tired, batteries get old- the extra stress sends the solenoid into e part heaven.