Peltier devices don't like hard on/off switching, for longest "life" pwm controls are the better choice.
With that said.. to simply make it "work" you could use ANY thermostat which would be usable for your home A/C Heating system provided they will operate at fridge temps (34F-40F) degrees (remote mount the sensor inside the cooler)AND provided you use a high current 12V relay between the T-stat and Peltier.
Keep in mind, Peltier devices are #1 EXTREMELY power hungry for the minimal amount of "cooling" they will give.. That cooler is going to easily draw 5A-8A at 12V (up to 96W)
#2 only cool about 30F - 40F degrees BELOW AMBIENT temps for a single stack.. That means if your air temp is 70F the cooling temp would be 30F - 40F, if your air temp is 90F you will get 50F - 60F which is entirely too hot for safe refrigeration..
Some folks however are undaunted by these draw backs and use them anyway..
Here is a "spec sheet"
Peltier Spec SheetBy the way, that Peltier "should" have a high temp thermal cutout wired in series.. That cutout may be bad which would cause the same symptoms of not cooling..
Additionally the outside fan may have failed, taking out the device in the process.
Personally (especially in your case) I would pull the heat sinks and cooling fins off the Peltier device and scrap the cooler... Just too much of a power hog and very little cooling power especially at high ambient temps.