Legal Question in Family Law in Kentucky

I cannot think of a legitimate way to get custody of his two daughters even if you had been married. The only way would be through some sort of agreed order with the contsent of their biological mother if she is alive. You might have a slim chance at arguing for some sort of timeshare but your standing in that is questionable.

If you two split up, then custody would be decided regarding your two children along with child support. You and he would have to divide the equity in the house since both of you are on the deed. If your name only is on the loan, then you are on the hook for that unless your file bankruptcy.

You have no cause of action for alimony (maintenance). There are a lot of protections that marriage affords two people that cohabitation does not since there is no common law marriage in Kentucky.

As for being stuck - well you always have choices, just none of them are good ones right now. One thing you might consider is confronting the affair and seeking help for the relationship with then end goal of seeing if marriage might be a possibility. The other is doing nothing which would be yucky and the third is ending it and seeing how it turns out.