Luckily I had an hold PC power supply I stole from somewhere, which has outputs varying from 33V AC to 5V DC, and of course 12V DC. The unit itself didn't have a switch, so I took the cover off to figure what wires actually switched the unit on and off... turns out it was the purple and grey wires which went to an isolated part of the power supply's internal board. They were connected together and the fan started spinning... so far so good!
Next I removed the old light, and the cabling with it. My original plan was to connect the wires that went to the old light to the power supply (via a male and female 240V plug). However once I wired it all up the old circuit didn't want to work. Nevermind.
Meanwhile I added two lighting strips to the cross-beams in the garage. This is pretty easy as the strips have 3M adhesive strips on them. They come with plugs pre-soldered on, so adding a second set was pretty easy. As you can see they put out a lot of light, pretty good considering they use much less energy than traditional incandescent lights.
I'll add more of these strips as I go - they only cost about $25 a pop. I'll probably eventually remove the halogen set I installed at the end of the garage as a second light source and replace them with LED strips as well.