I found, on the delightful treasure trove that is ebay, whole strings of surface mount LEDs on self-adhesive strips. Like these
At five metres long, this is good enough to go around the two long sides and one short side of a standard bed frame, underneath the highbed lighting up the play area underneath. Included on the strip with the LEDs are the right resistors to act as the current limiting element. This means that you can wire this up directly with a standard 12v PSU. To run the whole 5 metre length you need 24W PSU (capable of delivering 2A @12v ... P=VI, remember?)
If you don't want to use the whole length of the strip, or can't fit it into the space you need, it has marked cut points every 40mm (or 3 LEDs) along the whole length. Below you can see two sections with the cut mark in the middle.
Because we now have 300 individual LEDs spread out evenly, the light underneath is beautifully flat and shadow free. It's almost like florescent light, without the annoying flicker and breakable tubes.
Now, the next obvious step is to hook this up to an arduino and build a dimming control.