I would not have stayed, but Katie was really surprised by Jesse's admission and seemed to have only a few seconds to think about it. If the whole objective for every woman there for that entire evening was to get a rose so they could stay, it's logical that she would (and did) reflexively say she'd like to stay.
She may second guess that later, and perhaps we'll see a bit of confessional talk from her about that. After the rc she said that she was looking forward to showing him how great she is. I hope she can do that, but if she's not a first choice, it seems unlikely, in that hot field, that she'll be able to turn his head.
That said, Karen seems awfully weird, eh? When he said that she was really attractive, I thought, "yeah, in a Karen Black playing a crazy killer sort of way."