It had been two days since William Voge was publicly disgraced, and the 61-year-old globe-trotting lawyer couldn’t sleep.

His family, his colleagues and competitors, everyone it seemed, knew about the sexually explicit texts he exchanged last year with a woman he had never met in person, and still hasn’t. The fallout scandalized the legal profession and would cost Mr. Voge his job at Latham & Watkins LLP, one of the highest grossing law firms in the world, where he was chairman.

Mr....