1989 Honda Accord DX AC goes out

Kehin A/C is that a aftermarket unit? I have not seen one before. You say it was serviced ,what was done?
The switch you asked about sounds like a Low pressure cut out switch.When the freon gets low it kills power to the system.It is in the Low side of the system.
Or it could be a High pressure cut out switch that shuts the system down when the high side pressure gets to high.
Let me know what was done and we will go from there. I hope I have helped

It was the original unit for that model. Honda came out with a kit later that was Denso.

The reason for the service was that the AC mounting bracket broke so after replacing the bracket it was recharged 2-3 months ago and was working fine until last Wednesday.

The unit is full of refrigerant.

When I press the button the light does not come on around the button and the clutch does not engage.

Check the fuses under the dash and some times there are a few fuses outside under the hood too. One might say A/C. Does the fan still work ? If you do not have a test light " hooks to ground and a light comes on to show there is 12 volts there" buy one at wal-mart or a parts store 5.00 - 10.00 bucks max. Make sure the key switch is on if the fuse is good the test light will light up on both ends if it only lights on one end it is a bad fuse.
It is hard to tell by looking if a fuse is good or bad with out a test light.

Let me know.