Our refrigerator stopped working, compressor not kicking on.

There is a start capacitor near the compressor that may be bad. If the capacitor tests OK then the compressor is locked up. A service man can try to get it going with a kicker capacitor but there is no guarantee that that will work.

https://www.repairclinic.com is a good place to get troubleshooting ideas. Does the compressor make any sounds? There is usually some sort of starting relay or circuit to start the compressor and then switch to run mode. It is often inside a small box mounted on the compressor. If relay fails compressor will get very warm and make clicking or humming sound.

What sort of "humming" noise? Refrigeration systems rely upon vapor compression to operate -- meaning occasionally to maintain temp a compressor will kick in which can hum.

If the humming last 3-5 seconds then makes a faint click sound and stops humming for a minute or two. Then resumes the hum again it may be your compressor locked down. Stuck. Most likely it is just a bad start capacitor not kick starting the compressor. A $6 part.

Mine had the same problem, and I had a tech out today who determined it's the motor inside the freezer. It's (the fan) not running. It's a $100 part, but if your refrigerator is new enough, it's probably worth the investment to go ahead and have it fixed.

compressor or sealed system issue compressor is failing or a problem in the sealed syetem is putting a back pressure onto the comperssor causing the compressor to eventualy kick of on the internal overload in the compressor, cools down as you unplug the unit allowing the compressor to cool down resets the internal overload runs and repeats the process

have the unit checed by a quilified tech in ref sealed system repair

It sounds like the ice maker. If you don't have the water hooked up, and the ice sensor bar is in the down position, the ice maker assembly will attempt to make ice continuously.
Solution is easy: raise the ice sensor bar until it clicks in the up position. It will lock in this position, and the ice maker will not cycle anymore.

It will rattle slightly, this is normal. The ones that are burned out will sound like they have gravel in them, you can defininely tell something is wrong.
The problem with diagnosing Kenmore fridges with this problem is that there are different manufacturers. Yours is built by Frigidaire and probably has a Tecumseh compressor. We find more failed compressors with the Tecumsehs than we do with the models that have Embraco compressors. The models with Embraco's generally have more relay failures.

You should not hear the buzz/hum at startup. You will need to replace the compressor, which will be replaced at no charge to you if it is less than 5 yrs old.
If it is older than 5 yrs, you can have it replaced, but it will be almost as expensive as a new fridge. You can buy an aftermarket start device called a 3-in-1. This has a start capacitor built in to give it a little more boost on start up. I don't recommend using them unless it is a short term fix until the compressor can be replaced or if it is out of warranty and there is nothing to lose anyway.
Thanks/Mike

It looks like the Capacitor Start Relay is failing.  It is located either on the starter Relay or mounted on the side of the compressor compartment.  It sould have two wires on it, one orange and the other probably white.

compressor has failed. not a diy job