Use some windex and/or some plastic polish to get the pieces as clean as possible. The plastic polish helps remove any burn marks from the laser, and the windex will get all the fingerprints off. Everything will show, so take some time and get them as clean as you can.

Be sure to keep them facing the right way as you clean them.

Stack all the pieces and square them up as perfectly as possible. 

Use shiny aluminum foil tape to tape all the pieces together around the edges, leaving about 1/2" of plastic uncovered along the middle of one side. This is where the light strip is placed. The shiny aluminum tape acts as a mirror on the other edges, reflecting the light inwards and spreading it all through the piece.

Cut the LED strip to length, if you haven't already. Use clear packing tape to fix the NeoPixel strip to the part of the acrylic edge you left bare of tape. Line it up so the microcontroller is at the back of the design. 

Do NOT use the aluminum tape for this -- it's extremely conductive and you'll short out your light strip if it touches the copper pads on the back.

Once my light strip was covered with clear tape, I did add another piece of aluminum tape over the top to help reflect as much light inside as possible and to cover the end of the light strip.

I also added a little bit of aluminum tape as a narrow frame along the front edge.