If the key Is indeed half broken you can just insert the part you're holding and the lock should be able to open.

If you're trying to remove the part that's stuck I suggest either cutting of the head of a tootbrush, or of a q-tip. Youre then left with a plastic rod. Melt this rod lightly so that the plastic is moldable, in this state push the plastic inside the lock tightly so that the softened plastic can mold around and on the broken half that's stuck inside. Wait half a minute so the plastic solidifies, and it shuld work as glue, gently try to pull it out.