I've seen many back-lights monitors constructed, but none of them used their computers own power.
If you have a useless 5V cable in your PC, try doing the same.
Step 1: Materials
- LED strip (requires 12V input) // I thought about building my own step-up voltage regulator from this tutorial http://www.hobby-hour.com/electronics/lm2577-switching-regulator.php , but I figured out that buying one from ebay would be cheaper and the upside is that you can adjust it to a maximum of 35V which can be used for other projects.
-heat shrink wire
Step 2: Assembly
2- solder it together and to the Voltage regulator (make sure you give it enough wire lengh)
3- stick the led to the monitor
4- plug the voltage regulator to the EXTRA power source of your powersupply
PS: You can add a switch to turn off the light whenever you want.