To install the LEDs I tried simply drilling a hole slightly smaller than the LED and press fitting it. This didn't work because it actually pushed up the top laminations of the wood. So I used a slightly bigger bit that was the same size as the LED. This worked very well since the LED is actually have a very slight taper that I hadn't considered. 

I also found that if I drilled from the top, the drill would cause some delamination. So I tried from the bottom instead and if I went slow it actually created a fairly clean hole on the top. In order to locate the LEDs, I actually simply eyeballed the placement and used a spring-loaded center punch to start a hole on top. Then I drilled through with a small pilot drill, flipped over the board and finished each hole slowly with the full-size drill.

I used to very small wrench in order to get enough leverage to push them in. I don't think they will be falling out even without any glue, however I am giving this to a toddler and I could be very wrong if he decides to start whacking it with a hammer (which is not out of the question for him!)

Next I need to prototype all of the mounting by putting things in place with hot glue. If it all fits well then I can start wiring which may take me the rest of the week! There's a lot of wires in this thing...

Pics:

Test holes. Hole on the left delaminated.