The monitor finally gave up the ghost. It powers on, but no backlight. Fortunately there's a spare monitor, but I'll need to scrounge up a VGA/DVI cable or find an adapter (think I've got one somewhere). It's a smaller monitor, but it will do for now.
I think I'll take this monitor out to the bench and poke around inside to see if I can find anything easily repairable. A bit of Googling suggests the primary culprit is some bad capacitors on the power supply side of things, which should make it a repair well within my skill range.