Using a resistor to control brightness is a waste of power. You want to use a PWM circuit to control brightness. You may find more room and get a better result if you try edge lighting the screen with multiple surface mount LEDs (and that PWM controller).
Or you could leave the Kindle alone and get one of these for reading in the dark:
http://www.amazon.com/Lighted-Readin...ref=pd_sbs_a_7
Or this:
http://www.amazon.com/Glasses-Readin...6820437&sr=1-2