It's actually a very common solution to things being stuck, be it a broken plug or a annoying nut. Though it requires that you are rather careful with the application when doing it. The whole idea is to glue something on to that which you have a hard time grabbing to then use it to pull it out. Which does in fact, work very well when you are careful. I have done it many times over the years in everything from motor vehicles to computers with broken cables and plugs.
That being said, depending on materials, rubbing alcohol can be used to dissolve super glue. Nail polish remover can also work. Though be careful since they can also dissolve a lot of other stuff.
The correct solution from the original problem would have been to contact the school and say you got a plug stuck rather than making the problem worse.
Yep, it's a viable fix, but the plug being "caked in superglue" speaks to there being far too much glue and far too little finesse in the execution here.
Depending on the design of the port it might be possible to push out the plug from the inside, though that would require some disassembly of the laptop, and I'm not sure I'd trust the OP to do so carefully. It's also by no means guaranteed to be possible.
I'd try to remove the glue with some isopropyl alcohol or acetone/nail polish remover as a mentioned above - but BE CAREFUL. Use cotton swabs, apply a small amount (little enough to avoid dropping/running), wipe the glued area carefully (and nowhere else!) and be patient. Then I would hand in the laptop to the school and have them handle the repair.
I think jack is broken and OP tried to glue paper clip to it and pull it out.
Maybe use small drill bit to make a hole in stuck part and then use matching screw to pull it out.
Given that the OP failed at the "glue something on and pull it out" fix, I don't quite see how power tools is a responsible recommendation.