I (67m) discovered my wife (67f) is in another relationship. Advice?

A couple of months ago I discovered that my wife of 30 years (2 kids) is in a very odd relationship. The other man had nude and partially nude photos of her from 40 years ago on his Facebook page, as well as love poems. She has been sending him emails and phoning him almost daily telling him how much she loves him and wants to be with him, and talking about multiple rendezvous at a cabin.

Our marriage flips the routine stereotype, I'm the touchy-feely one and she is the strong silent type. It took weeks to get much information out of her. What I gradually learned is that she lived with this man in her 20's, heard last spring that he was dying of cancer, and contacted him. He lives a few hours away and is happily married with kids. The relationship is almost entirely fantasy fiction, they have only met once and his wife was there. They seem to living out a fantasy about what would have happened to their relationship if they had stayed together, me and his wife don't exist in this fiction. My wife, who is a very stable and rational person, insists that she has done nothing wrong and that this other relationship is of no threat to our relationship. She behaves entirely normally with me if I do so with her. She and the other man do not want to be together. The fictional relationship is not sexual, but is very physical (fictionally). After weeks of this I gave her an ultimatum, and she ignores it. She agreed to go to counseling with me, and then backed out so I went alone. We are both recently retired, comfortable financially, in very good health, and I thought we were set for a peaceful retirement.

After 2 months my feelings are still all over the map. I have always felt that the relationship was more important to me than to her, and I suspect that if I carry through with the ultimatum our marriage will be over. At 67 the dating scene is terrifying for me, and I don't really want to be on my own.