The sensor is probably a thermistor. Google "arduino thermistor" for thousands of tutorials.
The Peltier module could be burned out, so check that first. Use your multimeter to check the resistance, or power it up with any low voltage DC source, even flashlight batteries, to see if the module is pumping heat. The polarity does not matter.
And while you have that multimeter out, measure the resistance between the thermistor leads and report back.