It's possible, and has been done by lots of folks (google, and you'll see). However, it's not a trivial mod. You can't reuse the CCFL backlight electronics, so you have to figure out a way to obtain the requisite low voltage/high current for the LEDs. Then, you have to devise a way to connect and fit several LEDs in the small space allotted.

My suggestion is simply to buy yourself a replacement CCFL tube and be done with it. It'll last another 10 years, by which time you'll have moved on.