When the backlight does not stay on for more than a second, the problem is usually the backlight power supply board failure. Typical failures are blown capacitors, or transistors. A electronic repair shop should be able to fix it.

Look for bad caps as shown in my pictures here first: http://s807.photobucket.com/albums/yy352/budm/Sceptre%20X7G/

If you are going to DIY and have proper tools and know safety precaution then please read on:
Most common failures in the LCD monitors are bad capacitors (bulging top/seal or leaking) in the power supply (they should be replaced in a set), 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.

Basic LCD monitor troubleshooting guide:
http://www.fixya.com/support/r6150077-basic_lcd_monitors_troubleshooting

http://en.wikipedia.org/wiki/Capacitor_plague
Capacitors kit: http://lcdalternatives.auctivacommerce.com/ he can make you a set of caps for you.