This is pretty simple but you may have to break/open one of your connections. As I don't have my notes in front of me, I'm going by memory. That means I may have the green and white wire functions swapped herein. It'll be obvious which is which when you make a measurement.
First you'll need to remove the insulation on both of your newly made connections. IIRC the green wire is connected to the ground/common in the IC40. The white wire is the "sense" line and goes to an ADC in the IC40. With your thermistor still fully connected measure the voltage from the white wire (+ on DMM) to the green wire (- on DMM). It should read a positive voltage between 0 and 5 volts. If it's negative, my memory is bad and the wires functions are reversed from the above.
If your thermistor was good and the water temp at 77F, you'd measure 2.5V. Higher temps produce lower voltages, lower temps make higher voltages. Ideally 50F would produce 3.33V. A temp of 100F would produce 1.84V. If you measure a voltage inbetween these limits, your IC40 is likely good, at least from a temp measuring viewpoint.
If you measure 5V then either your sense wire (white) connection is open/broke or the thermistor is truly bad. If you measure 0V then either wire connection could be bad or your DMM isn't making good connections or the power supply inside the IC40 is dead. To go further you'll need to cut/open the sense connection (white wire). With the negative lead of the DMM still connected to the "good" green side, measure the voltage on the sense line connected to the IC40. It should read near 5V (mine read 4.96V). If it's 0V recheck your DMM connections. If they are good then I'd say your IC40 is dead Jim.
You can, if you wish, put the DMM in ohms mode and measure the resistance between the white wire connected to (only) the thermistor and the green wire. A good value depends on temperature of course but I'd expect between 11 kohms and 8 kohms. Be warned my thermistor measured in and out of range dependant on ... phases of the Moon.
If you read something other than 5V (+/- 0.2V) on the open circuit test for the sense line, then I'd cut open the ground wire too and redo the measurement, making doubly sure the DMM connections to the IC40 side wires are good.
If you come to the conclusion that the IC40 is bad then I'd do one last thing. Put a fixed 10k resistor in place of the thermistor and verify "wacky" operation persists. OTOH if days go by and everything works well, that points towards a bad replacement thermistor, as unlikely as it seems to be.
As to a test resistor, I picked one I had lying about. Nothing special is needed but do measure it's resistance with the DMM ahead of time.
ETA: at 86F the voltage (divider) measurement ideally should be 2.23V with a working thermistor. I've not calculated the tolerance range but I'll guess it to be less than +/- 0.1V.
