Now keep in mind that this was several years ago and I was really just dipping my toes into the whole DIY thing. There are a few things that I would do differently now. In fact, I plan to do a revision sometime soon. The upside, though, is that you can't really get much easier than this.
Step 1: What You'll Need
You'll need a power cable to plug into the Airport Express. If you've got a newer model than you've already got one.
Finally, you need an Airport Express. You have other options as well, which I'll cover near the end.
Step 2: Open It Up
Take a look at the situation in there. If it's an old speaker, it's probably pretty simple. You can see in the pictures that there's really just a speaker with some wires soldered to it.
Smarter people than me would probably tell you to to figure out what ohm the speaker is, they may use words like "polarity" or "polar bears", I don't know. Again, this was several years ago, but I'm pretty sure I just tinkered with it until it worked.
So, on to the next step.
Step 3: Connect the Speaker Wire
So connect the speaker wire to the 3.5mm audio cable. You could solder them and use heat shrink to protect them. That's what I'd do now.
Plug that jack into the stereo-to-mono adapter, and plug that into the Airport Express.
Step 4: "Mount" the Hardware and Test
The back panel of the speaker cabinet already had a notch to run a wire out, so that's where I ran the power cable.
Now would be a fantastic time to test this out. I'm not going to cover the configuration of an Airport Express. It's a piece of cake.
Let the sucker power up and try streaming audio to it. If it works, great! If not, try plugging something you know to work into the Airport Express — another speaker, some headphones, whatever. You may have a bad connection, especially if you went the electrical tape route.
So when it's ready to go, close it up.
Step 5: Rock Out
And yes — the name of my Airport Express is, "In Whatever Heinous Realm That It Squats". Deal with it.
Step 6: Some Notes
First, maybe you don't want to use an Airport Express. I got this bluetooth audio receiver from monoprice a while back. It's okay, though you have to re-pair with it every time.
Or you could do what I plan to do, which is loosely follow this very, very cool instructable and make an Airplay receiver using a Raspberry Pi.
There's also about 100 different, better ways to secure the Airport Express than tape.
Finally, check out the schematic that was inside the speaker. That's how you know this sucker is pretty old.