Stuck in cycle...

I have the same washing machine. I had the same problem you have. My problem was the timer. It just stops at that one position.
If you can slowly move it yourself and it starts to do something after you turn it a little bit then your timer is burn up in that one position.
You have to replace it. You cannot tear it apart and try to fix the timer. It is sealed by the factory.
UNPLUG THE ELECTRICAL CORD.
Take the back off of the panel at the top. Then you will see where the timer is bolted on. Remove the timer and take it to your nearest Kenmore dealer. Make sure you write down the serial and model number off of the back of the panel you removed. They will need it to get you the correct part.
You can sometime find some small companies that carry household appliances. They can order it for you at a cheaper price.

please check your timer . the timer motor may be bad, if it is so you have to replace it or rewind the coil

The problem you described, is fixable and to my opinion, you can fix it by yourself.

Check to see if the motor is running, then proceed as follows:
If the motor is running, your washer probably has a frozen pump pulley or a broken pump belt. To check the pulley, remove the pump from the washer and try to rotate the pulley manually. If it doesn't turn freely(frozen or stiff)replace it. If the pump belt is broken or looks quite worn, replace it--but be sure to check the pump pulley before you replace the belt.
If the motor isn't running, the lid switch may be defective. In this case, the washing machine can't spin and may not function at all. The switch is inside the washing machine main housing near the door frame. Often you have to raise or open the top or front of the washing machine to get to the switch. If it's defective, you need to replace it.

I just found this page because I was having the same problem with a coin-op machine in my basement (thus no settings to change). I resolved the problem by opening the lid and grabbing the center piece and shaking it back and forth as hard as I could.

Problem solved for today.