Is it her first time??
Some chicks are just don't know what to do when it comes to this.
Do not Wash the egg, it removes the coating it needs...however take a towel and ty to wipe off the big poops, cause you don't want anything affecting the chicks inside.
Next...Some broodies have to be forcefully removed from the nest a few times until they learn the routine. I never put food or water where they can reach it while they are on the nest. This only encourages them to stay on the nest until it is too late.
Make food and water available to her away from the nest, take her off the nest and put her near the water. You may have to do this a few days in a row before she gets the idea.
After a few days she will probably take time to eat and drink. Some times just moving around will cause them to poop before they get back to the nest.
As far as the kicking out...some hens know if an egg is bad, or she may not like the poop on them either....
Best Wishes,
Laura
Edit: spelling