Most common failures in the LCD monitors are bad capacitors (bulging top/seal or leaking) in the power supply, blown fuses, poor solder joints, failed inverter circuits (blown fuse, shorted transistors, shorted/open transformers), bad lamps (poor solder connections or worn out lamps). You will need to open it up and inspect the inside, see example of failed monitors to get some ideas what to look for: http://s807.photobucket.com/home/budm/allalbums
Post back what you see inside so we can guide you further and it will help out other people in the future also.
http://en.wikipedia.org/wiki/Capacitor_plague
http://www.badcaps.net