The Control Unit is a black cylinder with a silver label located just under the glove box. It is up behind the metal plate under the glove box and close to the door. It is held in place by a spring clip, and there is a five pin connector to it, although only 4 pins are wired.
Once I located and removed the control unit, I cut it open using a dremel saw. I cut about 1/4" below the top of the cylinder - the end where the wires go in. Condensation had corroded the bottom of the circuit board over the past 30 years to the extent that one soldered joint was completely loose. Ah-ah, thinks I. Fix that and all will be well. Not so. After a thorough clean up of the board and resoldering the bad connection, the symptoms were unchanged. Time to head to the electronics bench.
There is one IC and one transistor on the board. The IC is a CD4011 quad 2 input CMOS NAND gate. The output of each NAND gate should be very close to 0 volts when both inputs are at +12V. On my unit, the output was about 2V, had a significant oscillation and therefore was turning on the transistor enough to cause the warning light to glow.
A new CD4011 costs between $0.30 and $1.85 depending on where you buy it and how many you buy. Here on Vancouver Island I had to pay the $1.85 and wait a week to get it. But swapping it out cured the problem and hopefully I now have another 30 years expected life from my coolant level sensor. If it is 2045 and you now own my XJS TWR, this is how to fix the coolant level warning light.
I made a schematic of the control unit which is here.
And a photo of the control unit with devices labelled to match the schematic which is here.