Here are your parts
Replacement parts for KENMORE 79558823900 by Kenmore Elite | AppliancePartsPros.com

The Sears parts site has actual parts breakdowns.

The problem is that it is actually a LG unit but I do not have a cross reference list from Kenmore model numbers to LG model numbers.
I use a couple art numbers and they are also used in LSC269XX and LSC279XX models so from that came up with the following.

http://appliancedigest.c...rigerator-service-manual
You have to join the site to download the manual but it is free and this is a very good site for manuals and other info for the appliance repair DIYer.

Looks like it is fairly close to your unit but this is just an educated guess, perhaps you can find a better match on the site.

If the unit is running all the time you may have a defrost problem.
If the evaporator coils are frosted/iced over to the point where the evaporator fan cannot pull air through them then it is a defrost problem.

If there is just frost in one area of the evaporator coils after the compressor has run for a couple hours then you probably have a sealed system problem.
This will be an expensive repair but may be covered by your warranty. Check the manual.

If it is cycling then I would lean towards a thermistor (sensor) or a control board problem.