This type of issue is usually caused by the power supply. These are parts called capacitors that start failing. When the fail the power supply can't filter and regulate the output voltages correctly so the monitor does not power up. When you unplug and replug it in the power supply board restarts and will finally get going and will work until you power the unit off then you have to start all over again. The power supply board used in the VX2235 monitor is the Delta Electronics DAC-19M009 and it can be repaired easily by anyone that can solder a few replacement parts in place. We repair them in our shop all the time for this same issue. Here is a link to a repair guide we have posted that should get you back going again.
I hope it helps.
Wow, fantastic reply, thank you so much! If you could re-post the link for the repair guide that would be really helpful as it didn't seem to come through. I haven't found anyone here yet who knows how to do it, but maybe we can try ourselves, at home! Thank you for taking the time to post, much appreciated.