Drill holes (8mm) into the piece of plywood, as many as your LED-chain has. For my box I had a 24-light-chain.
Put the leds into the holes.
Drill a hole into the back of the BAS-box (14mm) and put the cable with the small plug through it.
Press the plywood as far as possible down to the bottom of the box. To keep it in place, fix it with some small screws at the sides of the box.
Reattach the screen of the box.
Disassemble the Nyttja-frame and pull out those little pieces of metal, that keep the back in place. The card back and the pieces of paper, that come with it, are not used here.
Lay it down, face down and put in the glass again. Put on it a photo of appropriate size or a sheet of printer-paper with some text or graphic on. I took a print on an overhead-projector-foil (Hewlett-Packard InkJet Transparency Film, HP3835A) with some piece of printer-paper behind it do get a diffuse light.
Put the box onto the frame, screen down.
Fix the frame to the box with picture-hangers, so it can be opened without tools.
Hang it to the wall, place it on a desk, put the plug into the wall: Let there be Light!