GT and Lobsterman are giving you good accurate information.
Lets talk some generalities here.
#1 it would be in your best interest to remove the entire fork from your bicycle. You have nice original paint to work around.
#2 remove the fork stem (only the 1 pivot bolt, loosen the pivot bolt lock nut first) from the fork assembly, you don't even need to remove the front wheel from the fork. So far, we have only spent five minutes of your time, tops.
#3 The key (or even the broken piece) cannot be removed unless the fork is in the "locked position". If you can ride the bike now, as in turning the fork, there's either a broken key in there right now, or someone stole some of the locking assembly from the fork. Since it's not easy to remove the parts, the most logical reason to the fork still turning, is you have a broken key still in the tumbler. It's easier to work on it on the bench. Take a flat blade screwdriver and turn the tumbler to "move the locking pin out", "before you try to remove the key part". Remember THE KEY WILL NOT COME OUT UNLESS THE FORK IS LOCKED, under normal circumstances.
#4 if you remove the fork stem, you will be able to easily see, and work on the fork stem. This is the easiest way to resolve your locking problem. It's a very cool mechanical part, after you take it apart. Even if you end up taking it to a "old school" locksmith, he will only want to have the fork stem. I've seen lots of good photos here on the CABE showing the step by step fork lock repair.
#5 Getting the broken piece out is the most important point. Don't damage the piece, the locksmith will need that broken piece to measure to make your new key blank. If you have the broken key piece, it's a easy cut for the locksmith, (read it will cost you less $$$$). If you don't have the piece, and don't have a key code number like most tumblers have stamped on them, he can still do a blank cut, but it costs more $$$.
John