Our Kenmore 110-24722300 has stopped working. It will fill, but then no response to the control dial. It has not spun off balance either. Help?

If it has filled with water and then stopped your problem could be the water level control. It is a switch that power comes into and first goes to the water solenoid valves and when the pressure on the diaphragm, via a plastic hose, gets high enough it switches over and sends power to a terminal on the timer and then to the motor and other devices. You will have to use a meter to check it out. Go buy a good digital multimeter from Home Depot, Lowe's, or Radio Shack, first. Here are the manuals you will need: https://dl.dropboxusercontent.com/u/8010163/Understanding%20automatic%20washer.pdf
https://dl.dropboxusercontent.com/u/8010163/How%20to%20use%20Test%20Equipment%20Service%20Manu al.pdf
https://dl.dropboxusercontent.com/u/8010163/Whirlpool-Kenmore%20washers.zip
https://dl.dropboxusercontent.com/u/8010163/How%20to%20Diagnose%20and%20Fix%20Everything%20Ele ctronic.pdf (This one has a chapter on using meters that is good.)
https://dl.dropboxusercontent.com/u/8010163/Basic%20Washing%20Machine%20Wiring.pdf
You will find it easy to fix your washer using a meter since you do not have to guess.