Let's start by preparing the wires for this project. You'll need six pieces of wire. I suggest 26AWG silicone coated stranded wires. Measure each wire to about 9cm in length.
Use wire strippers to remove about 3mm of insulation of the tips of each wire. Tin the exposed tips of each wire by heating it up with the soldering iron and applying a bit of solder.
Adafruit HUZZAH ESP8266
Secure the HUZZAH ESP8266 to a Panavise Jr. or helping third hands. Tin the USB, #12 and GND pins. Solder wires to these pins. They don't have to be color coded but it does help tell them apart.
Cut NeoPixel LED Strips
Next, you'll need to cut the NeoPixel LED strips into two pieces. Use the length of the enclosure to determine the right size (it should be a little longer than the diffuser).
If your using a high density NeoPixel strip (like the 144 LEDs per 1m), you may need to remove a "sacrificial" pixel. This is where you need to cut a pixel off so that you have more exposed pads to work with. Otherwise, you may find it difficult to apply solder to a pad thats cut in half.
Wire NeoPixel LED Strips
Now we need to solder the three wires from the Adafruit HUZZAH to one of the NeoPixel LED Strips.
But first, apply solder to each pin on both ends of the NeoPixel LED strips. While heating up the tinned pin with the tip of the soldering iron, quickly place a wire to solder it in place. Gnd to -negative, USB to +positive, and #12 to data in. Then, connect the remaining three wires to the end of the first NeoPixel strip.
Connect NeoPixel LED Strips
Now we need to connect the two NeoPixel strips together. Solder up the three wires from the first NeoPixel LED strip to the second. Make sure the marked arrows are flowing and not pointing towards each other. A pair of helping third hands will make it easier to solder.
Now is a good time to test your wiring before mounting it to the case.
Plug in a micro USB cable into the Adafruit HUZZAH and into your computer. You can upload the Arduino sketch or an example NeoPixel demo (like the simple sketch) to test the NeoPixel LED strips - just make sure to update the sketch with the correct number of pixels and pin 12.