President-elect Donald Trump rounded out his senior White House team on Thursday, elevating a set of trusted advisers who helped engineer his surprise election victory and bringing some of the rhythms and spirit of his unconventional campaign into the government he will soon lead.

Mr. Trump tapped Kellyanne Conway as counselor to the president, giving his former campaign manager a sweeping mandate to help make good on his election promises. Ms. Conway, who has become something of a celebrity in her own right, will be part...