I'm using 26awg (power & ground wires) or 30 awg silicone-coated wire (for data lines) for all steps.  This wire is super flexible, heat-resistant, easy to use, and very hard to break.  It makes the wiring on this project much easier using than traditional wire.  

Color-coding the wire is very helpful too, as the wiring can get a little crazy otherwise.  I'm going with the color-coding as I found it on my Dotstar strips.  Manufacturing sometimes changes these colors around, so notice if your wiring colors are different and adjust accordingly.

For me: Power (5v) is red, G is black, Clock is yellow and Data is green.  White is for the power switch.

Battery & Switch

Cut 2 white wires, one red wire and one black wire to just over a foot in length.  Solder the white wires to the two legs of the on/off switch.  Be sure to go through the tiny holes with your wire instead of just around -- a tight physical connection will be much stronger and more secure than just soldering.

Carefully cut the red wire coming from the battery, and splice your red wire on to the battery's power wire.  Once you've finished soldering and heat-shrinking, do the same with the black wire. 

DO NOT cut both these battery wires at once!  You might accidentally short the battery with your wire cutter, and could damage your battery or your body.  These battery-things are filled with powerful lightning. Respect them.

Place the switch on the bottom (non-wired) end of the battery.  Line it up so the white wires are on the opposite side of the cylinder as the battery wires. Secure the switch with a dab of hot glue.

Cut a piece of sparkly vinyl and wrap the battery, being sure the wires are spaced evenly down two sides and not crossing.  The battery with the LEDs will be a tight fit inside the tube, so get the vinyl and wires wrapped down as tightly as possible to make a slim, sparkly little package.

Hot glue the dowel to the other end of the battery.  Wrap the dowel and wires in another piece of sparkly vinyl, keeping the wires as tidy as possible and making sure they stay on opposite sides.

Wiring Everything Together

The Teensy needs to connect to 3 things:  the LED strips, the battery charger, and the IR sensor.  We'll solder 3 sets of wires into different areas on the Teensy, one for each of these connections.

LED Strips - Prep

Find the "in" end of the LED strips and put them at the opposite end from the battery.  Glue the strips down along the sides without wires.  Be sparing with the glue -- try not to add any bulk or your battery won't fit inside the tube.

Line the LEDs up as perfectly as you can.  A misalignment will make for a blurry picture when the poi are spun.  Take your time and get this right.

 LED Strips: Teensy Wiring

Put the poi assembly aside for now and grab your Teensy.  The wires to the LED strips will be soldered into the holes near the bottom of the Teensy, since that's where they'll line up inside the poi.  Since we have two LED strips, solder two wires each into the G (ground, black) and 3.3 v (power, red) holes along the Teensy's bottom edge, and two wires each into holes 11 (data, green) and 13 (clock, yellow).

Don't solder the other end of the wires to the LEDs just yet.  It's easier to get everything in place on the Teensy first.

Battery Charger - Prep

Clip the JST battery connector off the LiPoly backpack and clean up the + and - pads with your soldering iron. 

Flip the charger over and look at the back.  There's a little silvery patch.  Take your soldering iron and bridge the two pads here -- this will make your poi charge a lot faster.

To use the Teensy board with the Adafruit LiPoly Backpack (allowing USB charging), first two copper traces need to be cut: between the two pads next to the Teensy’s VUSB pin, and between the switch pads on the LiPoly Backpack (marked on back).

Battery Charger: Wiring

Add these three wires between the boards:

• LiPoly BAT to Teensy VIN/BAT+ (unmarked pin at corner)
• LiPoly G to Teensy GND
• LiPoly 5V to Teensy USB+ pin 

IR Sensor: Prep

Trim the leads on the IR sensor down to about half their length.  

Strip about 1/2 inch (yes, that's a lot!) of shielding from a blue, red, and black wire and slip a large piece of heat shrink onto each wire.  

As you're looking at the sensor with the bump facing you and the legs down, coil the blue wire around the left leg, the black wire around the center leg, and the red wire around the right leg.  Solder and secure with heat shrink.

IR Sensor: Wiring

Solder the black wire into the remaining G pin on the Teensy, the red wire into 3.3V, and the blue wire into pin 5.

Solder the LED Strips

Place the Teensy next to your LED strip assembly.  Trim all the remaining LED wires and solder:

• Teensy G > Dotstar G
• Teensy 3.3v > Dotstar VIN
• Teensy 11 > Dotstar Data
• Teensy 13 > Dotstar Clock

Note: some versions of the Dotstar LEDs have the pins laid out in a different order, so double check your strip layout before soldering.

Battery & Switch Connection

Connect the two white wires from the switch to the two power switch holes on the battery charger.

Finally, carefully solder the two battery wires onto the + and - pads on the battery charger.  If you see lights come on while you're soldering, click your switch once to turn the battery off.

Once all the wires are soldered into place, press your power button to turn the poi on.  Press the left and right buttons on the remote control to be sure the IR control is working. 

Do your best to resist the compelling urge to fling your proto-poi around through the air at this point to see the images.

Troubleshooting

If the LEDs don't come on, here are a few common mistakes to check for:

1. Plug the poi in to the USB cable and be sure the battery has some charge and didn't get shorted while you were soldering
2. Make sure your clock and data wires are not cross-wired to the opposite pads
3. Be sure the traces on the battery charger and Teensy have been carefully cut and you didn't scratch through anything critical
4. Try re-soldering the Dotstar wires using the pads on the back of the strip instead of the front.  I had more success soldering the data wires to the back of the strip and the power and ground wires to the front.
5. If you fry the first pixel in line, you can cut just that pixel off and try soldering to the next one in line.  The poi will still look fine with only 35 pixels on one side and 36 on the other, so don't worry!
6. Re-upload the code to your Teensy
7. Try uploading the Dotstar strandtest from the dotstar library to see if it's a soldering issue or a code / board issue