Thank you for your question. I understand how frustrating it can be to have your refrigerator not work properly and it sounds like you have made several diagnostic checks already.
A repetitive clicking sound from the control board usually means one thing. The control board is bad. Control boards will click when they turn the system on, this is normal. Repetitive clicking and no cooling is not. One of the relays on the control board is being energized and reenergized more rapidly than it should. A defective board will also cause the compressor to not run if the relay is bad. The control board part number for your unit is WR55X10942 and is available through [Sears PartsDirect | searspartsdirect.com?sid=MMLxexpertxmerch
I have also attached a wiring diagram for you to use when you get your new control board. If you have any further questions or concerns, please ask them here and I will do all that I can to assist you.
If you get to the point where you need to have a service technician complete this repair, you can schedule service through this link:
Sears Home Services