What should we do?

Hi so me and my fiancé have been together for 2 years. We also live together. She has a daughter and I do as well. She has been a lesbian for 12 years now. I am bisexual and I have been for 5 years now. We have been through a lot of ups and downs. Physical, mentally and emotionally! Through some breaks and break ups. Through the breakups we talked to other people. Nothing ever lasted over a week because we got back together. She has been talking with “A friend”( the same woman) for 7 months when we had our breakups.(Off and On supposedly). She also lied to me several times about this woman ever since the beginning. Anything that’s talked about she gets mad. She just wants me to get over the situation and leave it alone. So recently we were discussing how I felt about someone. (My child’s Father). I told her the honest truth and she said ok and that I should talk to him about it. So when I decided to talk to him about it she told me how hurt she was from hearing it. So she decided to leave me! The same night she left she went to “Talk” to the same woman from before to tell her how she felt. Since I told her the truth she was so angry she went and started dealing with the same girl again. We both done whatever we wanted to and she wanted to have a sit down with the 3 of us to let the (other woman) know that WE’RE OVER! It never happened. The woman also texted me disrespecting me from my fiancé’s phone. We spent more time apart and talked about things and we’re back together trying to fix things. She told me that she blocked her feelings for this woman since the first time! She claims that she’s just a friend and that she’s known her longer than me. So we both decided to dead what happened between us but I feel like she loves this other woman. She says she doesn’t she’s happy here with her family. I think if she wants this to work she should let the woman know that she is never going to do that again because going back and forth with this situation gone leave somebody hurt. I definitely don’t want that to be me! Am I doing too much by asking her to do that or should we break up? What should we do?