Kitchen Aid and Whirlpool brand refrigerator’s, compressor start devices have been known to fail. Having chased the clicking noise down to the back of the refrigerator and now realizing that the white box on the side of the compressor is causing the clicking noise and not cooling issue its time to replace it.
The device that starts the refrigerator is called the over load relay, the ceramic disk inside the relay will often fail causing a no cool condition.
The relay is inside of the plastic housing, having seen some of these get so hot they melt the compressor terminals I would not prolong the replacement.
By removing the rear cover you will gain access into the compressor compartment. The back panel comes off by removing the screws around the edge of the cardboard. Once removed place the rear panel of to the side. Located on the left the round black canister looking thing is the compressor, the fan on the right is known as the condenser fan motor. The overload relay is on the left hand side of the compressor by removing the white box off the side of the compressor and taking the wires off one at a time unless your model has a harness.
If you give the overload relay a little shake and get a rattle type of noise, sounds like glass is rolling around inside, this is the problem. Often refrigerator manufactures will change the design of the start device. Normally if there is a “different style” overload there will be instruction located in the appliance parts box.