Burke replaces Hall of Famer Grant Fuhr, who has been Phoenix's goalie coach since 2004 and will become director of goaltender development.
Gretzky -- also a minority owner of the team -- is awaiting a ruling by Judge Redfield T. Baum in the bankruptcy case of Coyotes owner Jerry Moyes.
King, a former Canadian Olympic and junior coach, has been an NHL head coach with Calgary and Columbus and an assistant coach with the Montreal Canadiens. He has spent the past five seasons coaching pro teams in Russia and Germany.
Burke, whose 19-year NHL career ended in 2007, played for the Coyotes from 1999-2004. He has been Phoenix's director of prospect development since March 2008.