Step 2: Make the Amp
It is a pretty straightforward circuit, very similar to any of the $5 amplifier circuits out there.
It revolves around the very useful 386 op amp. This is a great little IC with a few decent uses. There are virtually hundreds of amplifier circuits out there that can be built around one of these. Some of the better thought out circuits give surprisingly robust and clear sound.
I was going for a simple circuit, that delivered good sound, and this version does a fair job of that, so long as you keep the signal volume low (I.e the volume from your iPod or similar). Any sound system is only going to be as good as the speaker anyways, and with this type of speaker...well, not much better can be done.
The sound is decent considering the speaker, and the volume can be very high, however the speaker can't handle the output and crackles and distorts at high volume. A better speaker would certainly improve it and allow for a higher volume.
I honestly find that the volume is sufficiently loud even with the lower signal volume. A pot could be added across pins 1-8 to give control over the gain, with the 10uF capacitor, gain is set to the maximum. We tweak the sound a little on the way out to the speaker first with the R/C combination as a filter, and then smoothed with the final cap before the speaker. This last one doesn't make much difference if you are not planning on adding the LED Organ, but really improves the sound with the organ running off the output.
Solder yours up. Yours will probably be neater than mine.
Drill few holes in your speaker housing to let sound out and glue your speaker into the enclosure.
You could simply glue the speaker into the hat, but a box around a speaker helps immensely, even if the dimensions aren't engineered.