Anyone know what kind of switch needs a key to work?

That's a janitorial switch, sometimes called a key-operated or tamper-proof switch. The little slot in the middle takes a two-pronged key that looks like a tiny fork with two flat tines. You slide it in and turn it a quarter turn. They were standard in schools, offices, and commercial buildings where they didn't want people messing with lights or fans, wich fits with your paint-splattered chunky metal housing. It was probly installed when the building was a shop or office and someone repurposed the space into a bathroom. I ran into one controlling a fan in a converted storefront I worked on, and honestly half the time the key is long gone because nobody ever needed it. Good news is the wiring behind it is nothing special. It's just a single-pole switch, same two conductors as a regular switch, usally one hot feed and one switched leg going out to the fan. So swapping in a standard single switch is about as easy as it gets. Two screws, move the wires over, done. Just take a picture of how it's wired before you pull it apart so you can confirm nothing weird like the fan being fed from two places. Occasionally in older buildings you'll find somthing like a switched neutral or a shared circuit feeding a light and the fan off one switch, but that's rare enought that you'd notice it right away when you look. One thing to check before you buy anything: make sure there's only the one switch on that plate. If a light and the fan share the box, you'll want a double switch or you'll need to split it into two singles.

The bent box is worth sorting while you're in there. A mangled box with a gap around the switch can let a stray wire touch the metal edge, so bend the walls back out as best you can or pop in a new old-work box if it's really mangled. They make box shims too if the gap is just because the box sits too deep in the wall. Between the shims and a fresh plate it'll look fine.

If you'd rather keep it original, replacements with keys included are still sold for exactly this use case, so that's a valid option too. But for a house bathroom, a regular switch is what most people do.

Second the janitorial switch ID. I pulled three of these out of a converted dental office I was rewiring a couple years back, and the funny thing is the keys are basially universal. The one from the new replacement switch I bought fit the old original switches too. So before you buy anything, try flipping it with whatever key you can scrounge up first. Mine were wired exactly like a normal single pole, hot and switched leg, no surprises. Took me maybe 15 minutes per swap once the box was sorted. If the fan running 24/7 is bugging you, honestly just replace it and don't look back.

One thing I learned the hard way when I swapped a couple of these out: check the back of the old switch before you toss it, because alot of them had the conductors shoved into the little backstab holes instead of wrapped around the screw terminals, and those spring contacts loosen up over decades. If your fan has ever cut out intermittently, that was probably why. When you move the wires to the new switch, loop them clockwise under the screws and snug them down properly. Also, since the housing is metal and the box is metal, make sure the ground actually gets landed on the new switch, because the old commercial key switches frequently never had a ground terminal at all.

We had one of those in a little shop my uncle rented, and before anyone found a key someone just used a small flat screwdriver across the slot and turned it that way. Worked fine for years til he found the actual key in a junk drawer. So maybe try that before buying anything, could save you the whole job. Also curious, has your fan been running nonstop for ages now? Ours did that for a while and the little motor started making a humming noise, wonder if leaving them on all the time wears them out.