I'm doubting that you will bring the power consumption down that much, CFLs are also *fairly* efficient. I'm also doubting that you will end up with uniform light.
You'll have to take the monitor apart to see what will fit and can be mounted in place of the CFL tubes, and you'll need an alternate power source since the LEDs run at much lower voltage.
On the other hand, how much effort and cost is an old, 1080p monitor worth?