I don't know how to feel

I have been married to my husband for over 16 years. He went to a work conference in January and met a woman with whom he experienced a strong mutual connection. They began communicating often about work related matters and her son's behavioral issues following her divorce, which are similar to issues my husband experienced as a child. He initially hid this friendship from me because he did not think I would understand, and because they were confused about their feelings and the connection they have. They ultimately decided it is not romantic, but believe they are platonic soulmates and need to maintain a close friendship for some unknown reason. It took them a lot of talking to figure this out. I do not understand why. When I saw a picture of her on his phone as I glanced over his shoulder and then questioned him on it, he was very defensive, but told me who she was, what was going on, and how confused he is by it. I was extremely upset, both because he hid it from me, and because I am still suspicious that it is an emotional affair. He constantly tells me he loves me, is committed to me, and that he has no romantic interest in this woman, but feels a strong need to have this friendship. He also said that he feels we have lost our connection. I asked him to go to counseling with me. In the past he has been against counseling, so I was sure he would refuse, but he he agreed. Counseling has not been easy and I am struggling to trust him over this. He told her about it, and she suggested he pray about it and go to counseling. She is divorced and has said she wished her husband would have gone with her. She reached out to me by email to assure me that there is nothing between them but a close genuine friendship. I still cannot accept that. Complicating matters, this weekend he had another work conference spanning a weekend in her city. He asked me to accompany him and wanted me to meet her, but due to my work schedule, I was unable. I asked him not to see her, but he insisted on spending his free parts of the day yesterday and today with her, both alone and with her children present. In advance of this trip, she again reached out to me by email to assure me that there is nothing going on. Part of their time spent included attending her church, as she is quite religious and they had talked about their religion. My husband has a standing rule that he will never refuse an invitation to church from any of his friends. He has attended church services with other friends in the past. He says he cannot explain why he feels so close to this person, but that he has prayed about it, and gets a feeling they should remain friends even though he knows how much it upsets me. She wants to meet me and he wants us to be friends. I just can't seem to accept this friendship. I want to give him an ultimatum, but he will never respond well to that. What should I do?