Well, my phone is alive again. When I was messing around with the charger, I noticed that if I put it in halfway and sort of at a weird angle, the light that surrounds the charging port would light up and stay lit, like it does when you normally charge the phone. When the battery completely dies and you can't boot up, plugging in the charger properly doesn't make that light come on (it will flash briefly upon plugging in, but it won't stay lit).
So anyway, I figured I'll try and leave it in there halfway and see if that did anything. After half an hour or so, I hear that weird noise it makes when the battery is almost dead, looked over, and the boot animation is playing. It's now booted up completely and charging as normal (I plugged in the charger all the way).