sounds like you have a blown capacitor. Cheap monitors usually use crappy caps. If you take it apart, look for a bulging capacitor, or one with crap coming out of it.
This is what a capacitor looks like.
What you would need to do is solder off the bad one and re-solder a new one with equal voltage output. Make sure that you have the polarity correct. Match up the side with the strip to the same side it was or else it will just blow. That is if you decide you want to fix it. You also might want to look for surface mount fuses. Use the voltmeter to check for continuity across them. If you can't find anything wrong, a new monitor, even one with tv input, doesn't run too expensive anymore.