Your model number seems incomplete.
Kenmore numbers are usually 3 digits, a period and then an additional 8 digits.
I think you are missing the first 3.

Here are your parts
Parts for Kenmore Refrigerator 10671289102 - AppliancePartsPros.com

See the attachment for the tech sheet.

First ypou have to find out what is running and what is not.

Is the compressor running? You can usually hear it but also should be able to feel it vibrating.
Is the evaporator/freezer fan running?
Is the condenser/compressor fan running?

Do they run all the time except when the unit is doing a detfrost or are they cycling or not running at all?

I was originally thinking a thermostat, but from reading things I google up, I'm getting the feeling they either work or they don't, and don't do what mine is doing, any suggestions would be great I really don't want to buy a new unit.
[COLOR="Blue"]
A couple months back I used to always hear the fan starting up, now I don't. Which should be a good thing as from what I've read if the fan went I'd have more noticeable issues.[/COLOR]
No if the fan goes it will effect the temperatures ads the air flow will not be correct.

[COLOR="Blue"]I also hear a boiling noise from it almost daily which (from googling) seems to be refrigerant being pumped around?[/COLOR]
This does not sound good, normally you should not be able to hear the freon circulating.

If the fans and compressor are on all the time remove the rear cover in the freezer so you can inspect the evaporator coils.

If it is clogged solid with ice/frost you have a defrost problem.

If there is only frost in one area (usually where the freon enters the coils) or no frost you have a sealed system problem.

Do not do this check during or shortly after a defrost cycle or you may come to the wrong conclusion.