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Hello and thank you for using JustAnswer. I am very sorry that you are having this issue with your ex-girlfriend, I can understand how distressing this can be for you. Unfortunately rebuilding trust is not something that can be measured, which means there is no step by step manual on how to do it and there is no time frame that is associated with rebuilding trust. What you have to do is be completely transparent with her, show by your actions not your words, that you are being honest and forthcoming to her. Judging by your ex-girlfriends history with you and others, her issues surrounding mistrust and lying, it could take anywhere from a few months to indefinitely for her to fully trust you again. The choice to trust you is 100 percent on her, not on you, so she has to make the decision to trust you...all you can do in this time is show her that you are being true and transparent, what she does with that is entirely on her, not on you.
As for why she wants to remain friends with you, I cannot read her mind, so I can only speculate. It is possible that she wants to trust you again, but she wants to see what kind of effort you are going to make to earn back her trust and given that she wants you to have some relationship with her religion could mean that she is going to let faith/divine intervention "tell" her if she should trust you or not. Again, I am only speculating as I have never spoken to her nor met her.