On Kenmore dishwashers, there is a water heat cycle near the end of the wash cycle that must occur for the dishwasher to finish its work. The timer can move on to the next phase of the cycle only if the water is heated to a proper temperature.
This happens as a result of the water heating element in the dishwasher itself. If the dishwasher element is unable to heat the water, the cycle will not advance.
Even if the water gets warm enough, if the thermostat in the dishwasher is faulty, it will not be able to determine that the water is the proper temperature, and therefore your timer will not move forward. You can determine whether the thermostat is faulty by running the dishwasher through its normal cycle and opening it up as it reaches the end of the cycle.
If the water is hot, the dishwasher's thermostat is likely shot and needs to be replaced.
If the thermostat and the element are working properly, the most likely issue is that the timer itself is malfunctioning and sticking in place. If the element and thermostat work correctly, you will likely need to replace the timer.
The only solution for a failure of any one of these three system components is to replace whichever one is malfunctioning. Kenmore is a Sears brand; you can order parts from Sears Parts Direct.
Before attempting to replace any of these parts, always shut off the power supply to the dishwasher first to avoid electric shock.