Hi I'm John I will help you with the problem.
Does the motor keep going but no agitator.
The model number of the Kenmore washer is 110.92588210
I have replaced the cover and returned the washer to a difficult-access position, so I cannot see the motor at this moment.
That model has a capacitor a bad capacitor will cause that problem.
Replace that part it should fix the problem.
Your very welcome If you require more assistance let me know please.
Thanks and God bless.
yes its worth replacing and you know how .
The part number is XXXXXX and the price $132.84. At Sears.
Heres the video on how to replae that part.
No you have Accepted a great feed back you can do though.
Thanks and Cod Bless.