Agitator inside the drum is not moving during wash

Seems to be a common problem with Kenmore and Whirlpool machines. Follow this link to a good explanation of what's going on and how to fix it.

http://fixitnow.com/appliantology/wpdd_dogs.htm

Remove the agitator and inspect the plastic block on the shaft end. Is it split, or the interior splines stripped out? If so, replace the part. Leave the parts off and run the wash cycle--does the shaft turn back and forth? If not, then likely the coupling of the motor to the transmission has failed and needs to be replaced. Go to: http://www.fixya.com/support/r385252-replacing_motor_coupling_ w... by jsrock516. Give him a pat on the back if this helped.

whirlpool uses a coupler between the drive motor and transmission. and is the most common failure point. if the motor runs and the drum dont its this rubber coupling ..this costs about five dollars and is not too difficult to change if your mechanically inclined. I have always installed them as a precaution when preforming service on older wirlpools..or when they fail as seems to be your case.