I'm no expert on this, but have you checked for a tripped breaker? The power to the pressure switch should be continuous. The contacts should not be closed, unless it's on. You may need to replace the pressure switch.
Another thing to check is the line from the pressure tank to the switch. It may be clogged. Disconnect both ends and check it.
Have you checked the pressure tank pressure? Turn the pump off, drain the water pressure off, and check the tank pressure with a tire gauge. It should be two psi below your pump cut-on pressure, i.e. if your pump is set for 40-60 psi, which it apparently is, the air pressure in the pressure tank should be 38 psi. Air it up with a bicycle pump or portable air tank or compressor to 38 psi. If the tank won't hold the 38 psi without anything running, you need to repalce the pressure tank. The bladder is shot.
I think that you're probably going to at least have to replace the pressure switch and air up the pressure tank a bit to get it straight.