can a locksmith make a key from a broken key?

Absolutely. Keys can break inside a lock or outside a lock. If the key broke inside the lock a locksmith can remove the broken key allowing the lock to function again. In cases in which the key broke cleanly they can read the key cuts and make a new key. If for some reason a key broke outside a lock, they typically can just read the key and make a new one. If the locksmith is unable to read the key they can disassemble the lock in order to make a new key. One way to prevent keys from breaking is to not use the key as a handle. The key unlocks or locks the door. Turning the door knob or pressing on the lever is the way to open a door. Worn keys should be duplicated before they stop working or break.