My story is similar. I was married to my ex-wife for 14 years, and we had three kids. Yet this was when she started her secret, same-sex affair with a family friend, which I discovered in March 2019 after it had been ablaze for a year. Like you, I also wanted to keep it together.
With regard to your question about how the conversations went, I'll say "extraordinarily poorly". My ex-wife was in the process of shedding the fake life she had built in favor of her lesbian identity. I was an impediment to her moving on to her shiny new life (i.e., the life she should have pursued had she not been a coward), and she was downright hostile to me. Almost immediately after I discovered her affair, she said she wanted a divorce, loved her girlfriend more than me, wanted a trial separation to "figure things out", and identified a laundry list of my "faults" that drove her to have a same-sex affair. It was utter madness for me and the kids.
You are dealing with so many lows because it's a harrowing experience to watch the person who you loved and trusted the most willingly destroy the life you built with her, a life that you now know was fake. I dealt with the lows by sharing my story with family and friends (including the people in this forum), therapy, medical intervention, doing things for myself and my kids, and distancing myself from my ex-wife. I made it. You will, too.
I give this advice often, but it's far better for your kids to be from a broken home than to continue living in one. Your kids will rely upon you to model a proper loving adult relationship, which is something that you just can't do with your wife. You can't save a marriage by yourself. I promise that peace and happiness exist on the other side of this.