This is a great way to show off to your Mother-in-law that you can read.
I recommend building the KALLAX system in its entirety before altering any of the components. I located each of the shelves which would require strip lighting and marked them with blue masking tape.
Then, I disassembled each of the shelves and located where each of the assembly pins (dowels) enter the frame. (If those are destroyed, the shelf won’t stand up.)
I bought aluminum strips made specifically for L.E.D. ribbons and their complimentary diffuser plastic strips. The strips are aluminum U-channels (elementalLED.com) that go about 3/8″ deep and are about 3/4″ wide.
I used a router to create the channel where the aluminum fits (3 passes, since the particle board has a tendency to chip.)
The top/thicker shelf is hollow so please be extra careful. The edge (front 1.5 inches) of that shelf is solid-ish so that was enough for me to adhere the channel to.
I used a C-clamp as a guide rail and measured the depth directly from the aluminum channel (as opposed to the specification sheet) and secured the channels in the groove with white silicone. And of course wiped clean.
After creating the channels and securing the pieces into the shelves, I wired each row of lights into the LED dimmer (bought at a local vender, though home depot, and Ikea carry something similar.)
All of the wires go through one side. On that side, you can use one of the Home Depot cable management devices to keep the LED wires tight to the shelf.