I made sketches of the final setup. There will be a box in the rack that contains both the heater circuit and the temperature sensor boards. One of them is in the loop while the other isn't. Instead of having many cables leading to the can, there will only be these three, though they can be made into a single wire. It will be connected to the can through a D-9 connector. The second attachment is what will be inside of the box, with all the major wires and components labeled.

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Edit: I've canged the layout to (hopefully) make the labels easier to read. I've also added in a cable to the ADC that reads out the voltage across the 1 ohm resistor. I also attached the circuit diagrams for the heater circuit and the temperature sensors. The one for the heater circuit was made by Kevin and I used the same design, except I have LM7818 and LM7918, since the 15V ones were not available at the time I made the circuit. 

In addition, all the wires leading to the can will all be part of one bundle of wires (I didn't clearly indicate it as such). There will be a total of 6 wires: two are needed for the wire to supply power to the heater and will have a LEMO connector on the rack end and two are needed for each temperature sensor, which will be attached to the board directly on the rack end. 

Also, we don't need two voltage regulators for each temperature circuit. We can just have one of each of LM7815 and LM7915 to supply +/- 15V to the boards.