LED uniform lighting
I don't believe that uniform light intensity within that rectangular box is possible with only one LED. If I would have to do it, I would try a box made from special frosted plexiglas lighted by an array of LEDs outside with a distance of about 1 cm from each LED to the next LEDs and a distance of about 1 to 2 cm to the box. Of course each side of the box should be lighted with an array of LEDs.