I don't want to be Julius Malema about this, but now I'm not sure it's the battery anymore. Even after charging it fully (once fully charged, all the lights and indicators work again), it doesn't even make a clicking sound. Just one click when I hit the button, which sounds more like a short than a real click, and then nothing. Shouldn't even a shot battery be making a clicking sound from the solenoid/thingamabob?
Anyway, I'm going to clean the battery terminals and terminal connections, recharge the battery and replace the fuse and see how that goes. After that it's off to a professional, who will hopefully not break it any further (high hopes).
P.S. Can I test my battery myself with a multimeter, or does it need to go to a battery testing place?