This small refrigerator freezer model was sold at Lowes stores a few years ago and it seems that many of them are failing, including mine. The symptom was warm air being blown through the the inside. I did not get many good pictures of this so the instructable is mainly for information to someone looking to repair their same model refrigerator.
Step 1: The problem part
The reason the compressor does not run is the PTC (positive thermal coefficient) start/run relay has failed. After disassembling the back of the refrigerator and getting to the compressor you will find this relay buried under a cover.
In the back left is the capacitor.
The small white device with the blue, brown and white wires going in to it is the PTC relay (the failed part).
The round black cylindrical device in front of it is the overload switch.