My heart goes out to you. My husband had a similar scenario, except his wife was dating other guys and eventually left him for one of them. They stayed together for 10 years ostensibly for the kids, but the kids have since said they were glad when their parents split. They didn't feel part of a family when all the trauma was going on, with all the tension, rage and hurt, and all the extra people who were involved. Now they say they have two families.
I have to say I don't think your wife is considering the children at all; she's considering herself. She wants a cosy life with no real responsibilities for anyone else, and she wants you to pay for it. I'm going to blunt here - are you sure she's born again?
Only you can make the decision on whether to continue the marriage; as has been said, you have Biblical grounds for divorce. Put the kids' needs first; they WILL be aware of what's going on even if they can't articulate it. What will damage them less - being in a marriage gone rotten, or making a clean break. Is there any way you could gain custody if it came to a split? This marriage can be redeemed but only if both parties are prepared to work with God. Would she be willing to see a Christian counsellor with you?
I'll be praying longterm for you all.
Jules