Well, I'm back and I've had no luck whatsoever with the cleaning. Board is still dead and I've found no replacement parts online (unless there is some secret website that has it that I haven't found).
As a last resort, how likely would it be that a similar powerbutton board for a similar monitor model would work? Assuming that it's the same connector and I can physically plug it in to the motherboard, is it likely that it could turn on the monitor? Literally all I need it to turn on the display, I don't care about any of the other settings.