Easy enough to check for bad caps, just take it apart... but I usually see those causing 'No Power' issues.... Check warranty status first, then take it apart and check for obvious failure(blown/bulging/leaking caps)... If you can't find any visual problems with the caps(usually on the power supply) then I would just buy a new monitor. They are cheap enough, and I wouldn't want to spend time and money trying to fix an inverter/main board/back light problem.
Are you sure it's not just REALLY dim.... use a flashlight to see if you can still see picture... even if you do see picture, you will still be in the same boat though(repair wise).