I did something similar. I went with LED strips on the back of the monitor as I wasn't competing with a window behind the monitor but I think you would be ok. I also went with a straight 6500K white led strip as this is the recommended 'bias' lighting to make colors pop on your screen as well as reduce eye fatigue in darker environments. These lights were only $30 or so on Amazon. Powered by USB and I have a ton of USB AC adapters around the house. You could put the LEDs a few inches in from the edges of your monitor and be fine I'm sure.
I have mine hooked into a smart plug. Then using @stephack 's 'Boot Me Up Scotty' App and a Samsung Smartthings multi-purpose sensor, when I pull my chair out, the led strips on the back of the monitor turn on and the computer comes out of sleep mode. When the multi-purpose sensor is inactive for 20 minutes, the lights turn off.
