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.
Step 2: How to fix
A direct replacement relay is no longer available so a universal substitute is needed. I used a Supco brand URSC10 universal solid state start/run relay. It is rated for 1/20 to 1/2 hp and while the actual power of the compressor is unknown it works. This part is around $25 and has a wiring diagram on the packaging. Follow the diagram and use the existing capacitor in the refrigerator. The spade connectors will slip on the pins on the compressor.
Mine has been running perfectly for about three months now with this universal relay.