Find the center back of your hat and make a hole with an awl about 3/4" to the left of center, right along the base. The idea here is to get the first LED in the strip to be right in the center, so make the hole off-center enough to account for the wires and endcap of the strip. 

Also, double check and make sure you're starting in the back of the hat and not the front! I made this mistake and had to redo the whole thing -- the matrix effects will look better centered on the front, which means starting in the back.

Start winding the lights around the hat, making sure the lights are lining up in straight vertical lines, as best you can. My hat's sides are not perfectly vertical, so the bottom strip is a little bit off, but the rest are pretty good. I stopped before running into this same alignment problem at the top where it flares out a bit again.

It might be helpful to have some shims to slip in between the hat and the LEDs if your hat is just the wrong size to get the lights to line up.

Secure each strip in place with a bit of hot glue to hold it. The hot glue will hold but not super well so it's useful for protytping and adjusting the wire.

Once you get to the far end and you're happy with placement, secure it permanently to the hat using silicone glue or a physical connection like a zip tie or sewing thread.

Decide where you want to place your microphone. It needs to be on the outside of the hat to pick up enough sound. I put it on the front of the hat, so it will pick up the voice of the person I'm talking to, which is pretty fun.

Poke a hole for the Stemma connector and thread it to the inside, and then glue the microphone down securely.

Decide where you want to place your button. I put mine on the left side at the brim, slightly forward, right where many hats have feathers or decoration. 

Slip the button through the hole so it's on the outside. 

Plug everything in and verify it's all working. Once I was happy with placement, I glued the electronics securely to the "roof" of the hat. For the battery, I used velcro to stick it in place so it doesn't slide around, but so that I can remove it easily for charging.

To finish the encolsure I cut a circle of 1/2" EVA foam so that it fits snugly into the top of the hat. It covers everything except for the on/off switch, which I glued securely to the side of the hat about an inch and a half from the top. The EVA foam covers all the electronics but fits behind / above the switch, so it's easy to turn on and off. 

I used the upper portion of a pair of black opaque tights. They are already sewn into a circle and will stretch to cover the sides of the hat perfectly. 

I cut off the legs and used some FrayCheck to seal the cut-off end and keep the tights from getting runs when I stretched it. I also used my serger and sewing machine to make a narrow hem along the cut edge.

I glued the waistband of the tights to the top and bottom brim of the hat using E6000 glue.