I feel like telling the truth is the only answer. If the wife doesn't want to tell about her other relationships, It sounds like she doesn't have to, but at least you should be able to tell your adult daughter that you are in an open marriage and that you have another partner.

I am kinda leaning into the idea that you tell age appropriate things to your children about sex. And at 19 it is age appropriate that she can know about open marriages. I would also say it is age appropriate for 15 and 17 yr olds. You don't have to tell them anything more than that. I doubt any child wants the details beyond that anyway.