Busted lock

Since the key is in the slot you also might be able to use a small screwdriver and turn the tumbler and unlock the lock. I was able to do that one time with a broken off key still in the lock.

Roy Ken

If the key is all the way in the lock cylinder and has lifted the tumblers (pins) you should be able to turn the cylinder with a small screwdriver inserting it into the cylinder and twisting. After unlocking in you might be able to remove the entire lock. Most hardware stores should have a lock that will replace yours. But check to see if you can possibly push the key from the back side of the lock. I own a hardware store and handle the lock rekeying. If you get the old key piece out take it to the hardware store and they should be able to copy the key.
GOOD LUCK!

Google how to remove a broken key.

How to remove a broken key

Not necessarily, My wife did the same thing in our entry door lock. I was able to use a thin, stiff wire with a slight hook on the end to slip past the key and pull the broken end out. It took longer to find the right wire than it took to remove the key. Good luck!