Step one, shoot your teacher. There are times for stick soldering
practice. That wasn’t one of them. (speaking as a teacher, but
with my tongue mostly in cheek, mind you.) A better bet would have
been to either put slugs of solder on the inside of the piece, and
draw them through with the torch, or to pre-tin the joint, and melt
it in place. Either would have left you with less mess, and much less
risk of making a huge mess.
Meanwhile, depending on how detailed your surface is, files anywhere
you can get them to fit, in progressively finer grades. For
convoluted surfaces, I’m quite fond of snap-on disks. They’re the
little 1" dia. sandpaper disks with the square centered brass
grommets. You snap them onto special mandrels, and sand with them.
They die quick, but they’re incredibly useful, and come in all sorts
of grits. (AKA Moore’s disks.) The nice thing about them is that you
can flex them as they spin, to get into all sorts of concave areas,
and feather your cuts.
For really serious cutting into detailed areas, or delicate solder
removal, nothing beats the right engraving tool to fit the shape.
The problems are twofold however: knowing how to properly sharpen the
engraver, and then how to cut with it once you do. In the long run,
knowing how to use an engraver is incredibly useful for all sorts of
things, but it’s not a skill you can pick up in a hurry, or while
you’re desperate to finish something. It takes practice, not
desperation. So if you don’t already know how to use them, start
pondering and practicing. (the two most useful for solder cleanup are
flats of various widths, and an ongilette set up British style as a
Hope this helps, at least a little.