Kate did not do IVF. There were no eggs transferred.
Kate used Clomid and Intrauterine Insemination (they inject semen into her cervix) with both pregnancies. But, when she went to get pregnant with the tups, her ovaries became overstimulated with the Clomid and she had 8 ripe follicles. The Dr refused to inseminate her and hospitalized her. Apparently the Drs orders didnt matter to Kate and she had sex with Jon anyway and she was diagnosed pregnant 5 weeks later.
Their website's OUR STORY references the overstimulated ovary hospitalization and the fact that she was diagnosed pregnant a mere 5 weeks after the overstimulation. Furthermore, several times on the show Kate talks about being diagnosed pregnant with the tups and rubs Jons thigh and says, "You're not mad at me are you Honey?" An inference can be had that she had sex with him in an overstimulated state, and kinda tricked him to get her pregnant.
Reprinted with permission from Preesi.