Step 1: Open it up carefully
having said that, open the monitor carefully. There will probably be at least two or three weird overlapping wire connections (see picture) which you'll have to carefully unplug to get everything apart. Inside you'll find the LCD screen, a thick glass/plastic light diffusion panel, some thin plastic light diffusion sheets, and two CCFT (cold cathode flourescent lamps)backlights (at the top and bottom of the LCD screen).
Figure out which lamp doesn't work. This may be because the lamp itself is dead, or because an inverter output is dead (the lamps use AC which is converted via an inverter from the DC sent to the monitor). These lamps take high voltages, if you plug in the monitor to figure out if the lamp is broken be careful.