Have patience and a box of Band-Aids...
Seriously, you can make a tool out of a hacksaw blade that works without removing the lock. Grind most of the flat part of the blade away leaving a strip of teeth long enough to grab the broken piece of key. Use the "rear" end of the blade (that would be closest to the handle mounted in the saw) so that when you insert the tool in the lock the teeth are pointing back toward you. Insert the tool along the bumpy side of the key and then pull ir out. It should pull the broken key with it. Sometimes it takes a few tries, but I've pulled many a broken key with this old-timers' trick.