I've never had a broken egg in a hen before, but from all the poking around here at BYC forums, I know that she'll need a nice warm bath (very warm but comfortable to the inside of the wrist skin when you submerge your hand in the water is the right temperature). Are you certain it was a hard-shelled egg that broke? Could it have been a rubber egg (one without the hard shell coating) instead. Pray it was a rubber egg, it'll be easier to deal with. If it was a rubber egg, you can give her some scrambled eggs, hard boiled egg yolk, calcium supplement (from feed store or even from the human vitamins section at your grocery or pharmacy), or part of a TUMS (TM) antiacid tablet (main ingredient is calcium). This calcium will strengthen her contractions, and along with the warm bath, should help her expel any remaining membrane, albumen and yolk in her vagina and vent.
If she broke a fully formed, hard shelled egg inside, I would not induce contractions with calcium as I'd be very concerned with jagged (sharp) edges cutting her internally.
Again, I've NEVER had to deal with a broken egg in a hen, so the suggestions are things that I'd consider if she were my hen, until I received better advice from more experienced folks in these forums.
I'll be following this thread to learn more about how to handle this situation, just in case I ever have to deal with this situation with my flock.
I will keep you and your feathered family in my prayers.