My acrylic sheet is 1/4" thick. You can find this at your local Tap Plastics or order it online in the size you need. 

Use a rotary sander to sand both sides of your acrylic until it's translucent and fuzzy. The scratches from the sander are what catch the light so sand it thoroughly.

Trace out your design leaving around 1/2 to 3/4" of overlap. You want the lights to get close to the holes but NOT be actually visible through the holes. 

I used a separate piece for each wing. This kept me from having to buy and manage a GIANT piece of acrylic, and also meant I could run lights down the middle.

Cut the acrylic to shape. I used a bandsaw for this step. A laser cutter would work well too if you've got that available.

Now you can measure out your LED strip and attach it to the acrylic. Not much sticks to these silicone sleeves, but clear packing tape will! It sticks nicely to the acrylic too and is surprisingly strong and long-lasting and totally invisible from just a few feet away.

Face the lights inwards and center them along the edge, remembering to lay them out the way you'd envisioned in your design. Mine go up the center then around the outside edges and back down. Tape them into place.

Cut your strip at the end and seal it up with the heat-shrink-and-hot-glue trick.

Test your lights to be sure they're still working. Now it's time for the exciting part: mounting the acrylic pieces on the back of the bench.

I used L-shaped wooden brackets to hold the acrylic in place. I left the top edge open so the acrylic pieces can be removed and serviced in case of breakage (or to keep them clean in winter weather). This took a little bit of patience but the mounts feel very solid and also protect the acrylic from damage.