We had a very similar problem with my Kitchen Aid stainless side-by-side fridge PN: KSRG25FVMS01. Here is some information others may find useful in determining the failure.

Symptoms:
Both freezer and fridge were not getting cold, fan and lights stayed on, compressor would kick in for about 5 seconds and turn off. The compressor was very hot.

Troubleshooting:
I put a clamp-on meter on the line wire going to the Start Device (Grey Start Relay and Start Capacitor Assembly) and noticed the current draw was about 14A momentarily as it tried to kick on. The videos on You-Tube said 1A was normal! I feared it was a bad compressor. I pulled the Start Device off the compressor and disconnected the harness and measured resistance reading across all three pins. Start to Run = 10 Ohms, Start to Common = 6.7 Ohms, Common to Run = 3.5 Ohms - all reasonable. I then scratched some paint off the compressor and checked continuity from each of the three pins to GND (the bare metal area I just made). This verifies all three pins were not shorted to GND, they were all open circuits. Thus far everything looked OK. Now looking at the Start Device, after pulling the start capacitor off, I checked resistance from two spade input contacts (where the connector plugs into "Line and Neutral") to the 3 sockets that plug into the compressor. From Start to Neutral = 0.7 Ohms, Start to Line = Open circuit, Run to Neutral = 0.7 Ohms, Run to Line = Open Circuit, Common to Neutral = Open Circuit, Common to Neutral = 0.6 ohms. Once again all seemed OK. Measured the resistance across the cap, measured about 4Mega Ohms, which is reasonable. My meter did not measure capacitance. Finally, I partially plugged the cap back into the start device, just enough room to put my meter leads across the cap while in-circuit, and it measured a dead short. Coincidentally, it was shorted to the Neutral pin on the Start Device. Thats a problem!!! The cap is shorted to GND, never charging and delivering the boost of energy to the compressor to get its butt moving. I replaced the Start Device and now she runs like a champ. It costed me about ~$50, probably because it was Kitchen Aid. As a note, I do not recommend you replace the Start Device with a Universal 3-1 Start Cap, replace it with the same exact or OEM part you took out.

Just because you don't hear anything rattling around inside, doesn't mean it's not defective. Because I do electronics for a living, I wanted to find out exactly what failed. After opening it up, I discovered a very simple small circuit board inside with surface mount resistors, capacitors, and other solid state devices. Going through the circuit, I noticed the T1 device(PN: LT 8243 - I googled this part and could not find a datasheet) on the circuit board was shorted on all three of its pins. It's either a high power MOSFET or voltage regulator having a TO-220 style package. Nonetheless, he was the root cause. I hope this is a help to others!

I have Kenmore Coldspot Mod# 10653293201 approx 8 yrs old.
In that time this has happened twice:
Unit would stop cooling. Compressor fan would run but not compressor.
The LED lights on control panel top of refer section would be out.
I unplug the 120V plug wait 1 min, plug back in and presto, good for another
4 yrs. I am thinking bad OL/ start relay. But can they fail, then come back
to life and work for 4 yrs??

Thanks for the help.

ps can you tell be the OL/ start relay part# ?

Sorry for the multiple posts, I am new here :)

That would indicate an open winding in the compressor. You will need a new compressor. Thank you. Part number: AP4514660

Well, the freezer seems to be getting colder now(after a day or so, the bottom portion is freezing and it seems to be making its way up. I'm not sure about the refrigerator...we'll see what happens.