You are very much on the right track.
If the egg is still stuck and she will let you, take a cotton ball drenched in warm water and hold it against the cloaca where it meets the egg. Once the tissue seems to be moistened, gently, with a Q-tip soaked in KY Jelly (if available), oil if not, rim the edge around the cloaca (her 'vent') where it is stuck to the egg. if she struggles too much with this, just put her in a plastic tub/aquarium, with soft padding to sit on, and increase the room temp to 80-90 degrees F (27 C),. Increase the humidity to make the room almost steamy, and put her food and water close. Please give her fresh, pure water. You may give her a few drops of pedialyte/gatorade into the corner of her beak every few hours.
if she doesn't pass the egg in a few hours she needs to see a vet as this is a medical emergency. here is a site to help you find one: www.aav.org/vet-lookup . If this fails, start calling all the local hospitals and BEG.
Your bird is depending on you.
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