O wise biblio! Yes, conveniently/fortunately/providentially are good here. (But personally, I'd omit the adverb....)Inasmuch as the person is about to do something that does not make sense, the sentence makes sense.
One idiomatic version would be: "He was looking for someone to have an affair with, and his sister's friend conveniently/fortunately/providentially turned up."
I should warn you, neither providentially nor opportunely are very common in everyday conversation or, indeed, in most writing. The adjective forms of these words are more common, on the other hand.
Devereux"Ay, in all things and /all places/; eh, Count?" answered the Regent,
learned how you came so opportunely to our assistance that night. /Dieu
me damne/! but it reminds me of the old story of the two sisters meeting
at a gallant's house. 'Oh, Sister, how came /you/ here?' said one, in
virtuous amazement. '/Ciel! ma soeur/!' cries the other; 'what brought