I just started reading their book, and in it they state that Kate was showing 4 eggs that month. They considered postponing the insemination, but they thought through the possibilities. They thought it highly unlikely that 3 would be viable, but decided that if it happened, they could handle it in their financial and living situation, though it would be tough, and that 4 was like a one in a million shot, so they decided to go through with it. They said that the injections were painful, and I imagine that they were pretty expensive too and not covered by insurance, so They probably felt like waiting another 2 months, having more injections and paying for them over a one in a million shot was not necessary...and then it turned out that there were some eggs that did not show originally, hidden like a previous poster mentioned.
Also, I'm giving Kate the benefit of the doubt here, but I don't think that she believed she was entitled to the nurse for those reasons. I think that she was just a mother willing to do or say whatever she could to provide for her children. I think she felt that her children were in need of the nurse and that they would be unsafe without her, so Kate said whatever she needed to to get it for them. While I think that lying is wrong and immoral, and that it definitely comes back to bite her in many online discussions, I can't say that I respect her any less for doing what she felt was in the best interests of her children.