First , I hope your mother in law is doing well. I know you have had a lot of advice on this compressor. Im just trying to give my two cents. If nothing is making sense or you are not comfortable doing this, please hire this out to someone to trouble shoot this. I believe in a earlier post you said if you manually pushed the contactor in, the motor ran. So that is telling the motor is ok and you have power to the contactor and the power wiring in also ok. What does the pressure gauge read, is there enough air in the tank keeping it from running, in not then the problem is in the control. There are 4 components here.
The coil
The overload relay
The pressure switch
The wiring connecting all of these
Please make sure power is off to the contactor before doing any of this, don't take anyone's word, check yourself. If you don't know how or don't feel comfortable, stop here and hire this out. You have to have a basic knowledge of electrical trouble shooting to check these components. Check the coil voltage by looking on the actual coil, not paperwork. It should say 240 volts, not 24,120, 480. If it says any of these other voltages, here is the problem. If it is 240, there are 2 wires on that coil, remove at least one. Hopefully you have a multimeter and are familiar with it. Put the dial on ohms, touch the leads together, it should read zero and might even beep. now that you tested the meter, put the leads on the coil terminals, It should show some resistance, probably low, but some. If the display shows OL, then your coil is burnt open. Something I should have mentioned, take a picture with your phone in case you forget where the wires go. If the coil checks ok, lets check the overload relay. take one wire off the terminals, making sure that it is wired to the normally closed set of contacts if there are both. Put your probes on these, your meter should read zero, if not, try pressing the reset, if it says OL, then the problem is the overload relay. Now the pressure switch, remove one wire, this also should be wire to the normally closed contacts. Read this with your meter, If it reads zero, it is fine. If it reads OL, the switch is open. Check for air in the tank, if none, the switch is bad or wired wrong. Only thing left is control wiring. without knowing your skill the easiest way for me to tell you is remove one wire at a time and check continuity with the meter, reads zero the wire is good, reads OL the wire is bad. Especially check those stay con terminals on the pressure switch wiring. Do all of this one wire at a time then put it back before taking another one off. If all this checks out good, then it is not wired right. There should be a control wire from line one from the power going to one terminal of the coil. There should be a control wire from line two of the power to the overload relay, then second wire from the overload relay should go to the pressure switch, then the second wire from the pressure switch should go to the other terminal on the coil. Sorry for a long post, hope this helps