On a Sunday afternoon not long after he was elected vice president, Joe Biden recalls, he learned a lesson about his new station in life. Biden was finishing up a call with his new chief of staff, Ron Klain, when he dropped the news that his friend Ted Kaufman would be filling his Senate seat. “Did you tell the president?” asked Klain, who, as an aide to Al Gore, had learned something about vice presidents and their place. “Why the hell should I tell the president?” Biden remembers thinking. After all, for 36 years, Biden had been his own man,...