This poor gal had a chicken emergency last week, with an egg bound in the vent that did not contain an egg but a mess of shell and poop. With gloves on and armed with warm water I gave her a good soak and waited to pull remaining items from her bum. It could have only been 24 hours but she had a bulging red underbelly. She still had an appetite and attitude so only time would tell if she would kick the bucket or not. The feathers fell out of the bulging area but within a day the feathers where coming back along with regular droppings. The redness was going down along with the size. Here she is a week later and still no eggs with shells but better than dead. Here is official advice on how to handle this situation:
"An egg travels down the oviduct, pointed end first. When it gets to the urogenital chamber of the cloaca, it turns end for end so it will come out large end first. Sometimes an egg gets stuck. The egg may be too large (for example, if a pullet lays unusually large eggs), or a disease may cause the oviduct to swell or its muscles to become partially paralyzed. The stuck egg then causes future eggs to accumulate behind it, distending the hen's abdomen. Unless you can get things moving again, the hen will die.Lubricate a forefinger with mineral oil or KY Jelly and insert it into the vent. With your other hand push gently against the hen's abdomen to force the egg toward the vent. If you can see the egg, but it is too big to pass through the vent, puncture the shell and remove it in pieces (with great care not to let a sharp shard injure the hen). Rinse the cloaca with hydrogen peroxide.
If the egg was stuck in the cloaca too long, cloacal tissue may protrude through the vent. In that case, protect the hen from cannibalism by isolating her until her muscle tone is back to normal."