What a great story! Thank you for sharing. I'm not surprised she stays, your place is her home. Give them food, water and shelter and you can't get rid of them 
Pigeons do co parent so keeping food and water next to her now is a good idea. They also feed their babies in the nest until they're feathered out and learn to fly so I'm not sure if chicken eggs would be your best option. I would also be concerned about her ability to manage it all by herself. In my opinion it would be best to let her sit but not give her any fertile eggs. Some day she may find a mate.
Best of luck to you and Egberta!

Pigeons do co parent so keeping food and water next to her now is a good idea. They also feed their babies in the nest until they're feathered out and learn to fly so I'm not sure if chicken eggs would be your best option. I would also be concerned about her ability to manage it all by herself. In my opinion it would be best to let her sit but not give her any fertile eggs. Some day she may find a mate.
Best of luck to you and Egberta!