My, your friend is very open about her relationship, isn't she.
I wonder why this couple are engaged as there seems no prospect of them marrying, and although they live together it is not a full time commitment. Is she just becoming a nursemaid? Why would she need more evidence to catch him out; isn't the evidence she has discovered by reading his personal emails enough, or doesn't she want him to know this?
If it were me, I would ask him outright what he is doing, then retire to my own home and leave him to stew while I thought hard about what I wanted from this relationship, and consider whether it is worth preserving.