Welp, my first time soldering, my first time doing anything of note with electronics, and it's working - all thanks to you guys on this forum!
I managed to solder/tear off the busted fuse and solder on the new one. This took a bit of internet searching and luck on my part, ripping a standard plug apart to get at the brown wire - to reuse for connecting the fuse. With an overabundant amount of solder, I made a thing.
I then wrapped my mess of a job in electronic tape, screwed the back on and powered it up.
Success!