Parallel strings of 4 or 5 LEDs in series with anodes powered from 12V and with a single PWM controlled N-channel MOSFET to sink current from cathode/current limit resistor would seem sufficient. 5mW per square millimeter is not much. A 2.5V LED with 20mA of current flow uses 50mW of power. A 5mm LED has surface area of 19.6sq mm.
So you could say 50mW/19.6mm^2 = 2.6mW/mm^2?
So a top & bottom layer would provide 5.2mW/mm^2.
Do you mean mW/mm^2, or mcd/mm^2?
Many high efficiency LEDs have mcd rating in the thousands/LED:
If you had a grid of these, you'd have pretty high mcd/mm^2 also.