What did Jung's patient find out?
Jung does not tell us what his patient found.

He circumscribes the woman as appearing "in very unflattering guise" and "connected [...] with all kinds of scandalous stories". What kind of unfavourable scandal she was finally discovered to be involved in, Jung does not disclose. We can imagine anything from premarital sex to secretly working as a prostitute. What Jung's patient at the time of Jung's writing might have considered "scandalous" could well be seen as completely innocent and commonplace nowadays (e.g. premarital sex). Times change.

But the important aspect of this section is not what the woman did, but that the patient unconsciously had come to an insight that his conscious mind had not yet made, and that his unconscious informed him of this insight in his dreams.