The following day, she still hadn't laid an egg. We gave her one more day, but the poor thing was looking so puny and sick that we made the decision to put her out of her misery. We then decided to do a quick autopsy to see if we were right that she was egg bound. It turns out the poor thing was full of this huge yellow mass. After doing a little research, it looks like we had an "internal layer."
From the Merck Manual:
In these hens, partially or fully formed eggs are found in the abdominal cavity. Such eggs reach the cavity by reverse peristalsis of the oviduct. If they have no shell, they are often misshapen because of partial or complete absorption of the contents. Frequently, only empty shell membranes are present. No control or treatment is known. This condition is related to erratic ovulation and defective egg syndrome.
So the mass I was feeling was technically an egg (really a lot of eggs) but all the soaking in the world would not have helped our hen. Once a chicken starts laying internally, the kindest thing to do is to put them down before they die from infection. Hopefully, if we ever see this again, we will recognize it sooner and take the necessary step to keep our ladies from suffering.