Do not use a white LED, use an orange LED instead, 3mm, 2 in series each side turned to light the glass of the CRT. Much better illumination and a much better contrast with the blue white of the CRT, plus the LED's do not leave a flood of activated phosphor on the CRT at high brightness. Then you can use any convenient power rail around 12V that has 20mA of current capacity, and a 10k variable resistor across the 12v rail, feeding the base of a NPN transistor (PN2222A) via a 1k resistor, and collector to 12V, and a 1k 0.5W resistor from emitter to the 4 LED's in series. Gives a good brightness range from off to way too bright.
I did the same for a USB light, though it spent most of the time running at 1mA, as the full 20mA of current through the recycled white backlight LED's I used ( amazing how bright those suckers are these days at 20mA, when they run in the panel at 50-100mA) was basically daylight.