Monitor on/off problem

You would probably hear a relay click if there was one. It probably doesn't even have a solid state switch. It probably just goes into sleep mode instead of actually disconnecting power.

I'd also think it is just a momentary switch and the PSU main output is started / stopped electronically.

Check the switch with a meter while power is totally isolated.
You may be able to clean it by putting a few drops of isopropyl alcohol in it and working the mechanism for a while..

Also possible the switches pins have come loose in the PCB. Re-soldering them might help.
Those momentary switches are very cheap to replace, might be better to do that.

This push button switch can not be open is seal switch .
It is momentary type switch push on /push off.
might psu circuit,don't know how that circuit look like.
mrel

Please clarify, a momentary switch and a push on/push off switch are completely different. But if it's one of the switches on the board at the front in your photo it is very easy to repair, because the fault you describe relates to the switch itself rather than anything it's connected to. To test the theory you can track solder a couple of wires to the pins at the back of the switch and turn the monitor on and off by shorting them together.