Got it to work. A cable was broken and the AD586 also broke at the same time so it took a while to find the problem. I had to create a makeshift cable out of three parts so once I replace it for an actual cable, it will be good to go for a test.
Today, I stuck on the sensors to a metal block using a flag, rubber bands, and some thermal paste (1st attachment). I then wrapped the whole thing in about 4 layers of insulation and a lot of tape (2nd attachment). The only things leading out of the box were the three connections to the sensors and a thermometer. I then connected the wires to their respective places on the board of the sensor. To get the readings out we would need to use an ADC. Gautam and I checked to make sure the ADC we have inside the lab goes from -10V to 10V so that it would be able to measure the 3V value the sensor typically measures. We then tried to connect all three sensors to a DC source simultaneously, but unfortunately one of them seems to have disconnected somewhere during the process, as it only showed 1.2V instead of 3V. I plan to fix this tomorrow morning so that we can hopefully set this up soon.
I took off the AD590 and attached it to two long wires leading out from the board. This will allow us to attach the sensor to a metal block and not have to stick the whole board to it. I have also completed three identical copies of this and it's pretty much ready to be tested. According to Craig and Andrew's elog here, the sensor is very noisy and they added in a low pass filter to fix that, so that's something to consider for the final version of the circuit. I'll test what I have so far and see how that goes. We still need to figure out how to get readings from the sensors.
To attach the sensor to the metal block, I'll use some thermal paste and fasteners. I'll also put a thermometer on the block to record the actual temperature. I'll then wrap it in some insulation we have in the lab and have only some wires leading out of it to make measurements. I'll leave this setup overnight and record the outputs for about a full day. The fluctuations between the sensors will then indicate the noise of each individual sensor.