So I tried clicking the fire button a few times and that helped a little - I can now navigate around the menus and select games using my joystick. Unfortunately, that's as good as it gets. When I actually try using it in game, nothing happens. I've tried it on a few games (Zool, Superfrog, James Pond, Mercenary, Diamond Mine, Into the Eagles Nest) with no luck.
It's not really a huge deal, I do have a USB joystick I can use. It's just the Speedking was always my favourite as it's easier to hold and doesn't make my wonky thumb hurt after a few hours gaming.