Product review (5 stars): Amaze and impress your friends and colleagues...
Why is it that the ability to fix a fridge that kept freezing up here in the office is more impressive to my colleagues than the work I actually get paid to do? Bloody engineers... Regardless, it rescued a perfectly good 4 year old fridge from being tossed out.

So, three things* that can go wrong - the heating element can be burned out, the thermostat can have failed, or the timer / control board may have failed. Once you have determined which it is, then order the replacement part.

*Actually, there are more than three things, but these are the most typical fail points if your cooling system is functioning properly and your door seal is in good shape.

>> Remember to unplug the fridge before doing any diagnostics unless you like getting shocked and possibly permanently damaged or killed. <<

The heating element is a simple ohm check with between 30 and 50 ohms typical of a good element. Infinite resistance means a dead coil.
The thermostat is also simple to check. Unclip it  from the coils and use a continuity tester and a cup of hot and one of ice water - It should switch open after soaking for a minute in the hot water and switch closed after a minute soak in the cold. If it doesn't, replace it.
The timer / controller is a bit more tricky, but if the above two things are good, then you'll need to do more research than this review can provide.

I suggest starting your diagnostics work in the evening, because the coils will probably be a massive chunk of ice that will need to thaw out overnight. Also, the drip tray will probably not hold all the resulting water, so a few strategically placed towels (vinyl flooring) or a baking sheet placed under the tray (wood flooring) may keep you from discovering a large puddle in the morning. You'll need to empty your freezer, pull out all the brackets, shelves and ice maker, to get to and remove the back sheet covering the evaporator coils* in the freezer. For this, you'll probably need a 1/4" socket, and slot and Philips screw drivers. If you are lucky, you'll be able to get to the heating element (at the bottom of the coils) and the wires running to everything will not be embedded in the ice. If not, put all your frozen food in your cooler chest along with a bag of ice, prop the freezer door open and go to bed.

* Have a care not to nick or puncture those coils, so don't go poking with an ice pick or screw driver to "speed things up." If you do, things just got really expensive. Better to let the ice melt naturally with the freezer door propped open. Use a hair dryer on low setting if you are in a hurry.

So, you have access and have done your diagnostics and determined it is the thermostat,  Congratulations, you've already done the hard work. It took you longer to get to the thermostat than it will to replace it. Now, put the coil cover/back sheet in place and load all your food back into the freeze, then plug the fridge back in - it will keep things cold while you wait for the thermostat you just ordered to arrive. Once it arrives, you'll need to have two butt end crimp connectors or wire nuts handy along with a wire nipper/stripper. Simply cut the wires close to the old thermostat, strip the the insulation back and connect the new thermostat in place of the old one. Clip it back to the coils, tuck the wires out of the way of the fan and put everything back in place.

See, you too can impress those around you with your mad fridge fixing skills.