The dishwasher runs the wash cycle continuously, never moving to the next cycle. I pressed the cancel button and the dishwasher does drain. The soap dispenser did open and dispense the soap. Which part do I need to replace?

Serial # FW3907913.

Spebola, a couple of preliminary checks. Start the dishwasher in a cycle, allow the dishwasher to fill and the motor to start and run a minute. Unlatch and open the door, remove the lower rack and check the water level in the bottom of the tub, it should be within 1/8 to 1/4 inches of the heater element. Also check the water temperature, it should be at least 120 Dgrs.F. at this point.
The K/A dishwasher has thermostats and sensors that control the time in the control. if the water temperature is too low, or there's not enough water in the tub, the control can "stall" until the proper temperature is reached, causing the wash cycle to run extended periods of time. There are other possibilities, but start with the basics first

Spebola, No, This does not mean there is an issue in the heater element.
You can check the element resistance with a multi meter, it should have a resistance of 20 to 30 ohms. But the water temperature is more important at this point. The lower the water temperature in the tub, the longer the "thermal hold" cycle lasts. If the water temperature is 100 dgrs. and it has to heat it to 140 dgrs. you're looking at a "stall" time of 40 to 50 minutes, per thermal hold cycle. You could be adding as much as a couple of hours to the wash cycle.

We've attached a copy of the tech data sheet, so you'll have a better understanding of the dishwasher operation and can conduct the diagnostic tests, and check circuit resistances if needed.