I fixed my two old Chimei monitors about 18 months ago. In both cases all the black caps in the power board bulging so grand total of $15 or so @ Jaycar and 30 minutes or so work each fixed it.
Although when I say fixed it's not 100% - there seems to be some AC induced flicker that I don't think was there before in the fluoro backlight although don't they run on AC anyway?
In your case my guess would be either fuse or powerboard has gone bad, or it might be something as simple as the power microswitch no longer functioning.
Opening the thing up shouldn't be too hard, if it's not screwed then it's probably a clamshell type thing with strategic points to put pressure on or insert something thin into to open up.
The beauty of most monitors and TVs these days is that the commonly failing stuff is just easy to get to. Literally 3 minutes or less and you have the power board removed without disturbing much else.
No doubt you should be able to test if the microswitch is bad by a comination of overriding it by jumpering as well as continuity testing while pressing the button multiple times.
Footnote - some monitors AFAIK have that HDMI auto power-on function although it's probably an option you'd have had to set earlier.
Also there's the plug/play detection that will often work on a monitor that's not even powered up though I think that's because the HDMI is supplying power for that to happen.
Edited by Rybags, 07 November 2016 - 03:13 PM.