It usually pays to think simple first. Squirt or dribble some penetrating oil or WD-40 into the chamber, and let it soak into the case-chamber gap.
I would try jamming a cleaning rod with .45 or .50 cal bronze bristle brush into the chamber and the broken case. Twist, and it should come right out. It it is hard to twist, using a long cleaning rod from the muzzle end, tap the case-brush assembly and it should come right out.