The Stanley Cup champions announced the moves to keep coach Darryl Sutter's staff intact Wednesday.
Stevens is a former head coach in Philadelphia. He briefly served as the Kings' interim head coach before Sutter was hired during their 2011-12 championship season.
Stevens had been mentioned as a candidate for off-season job openings in Vancouver, Pittsburgh and Carolina.
Payne is the former St. Louis coach who joined the Kings' staff in 2012 after their first title.
Ranford played parts of 15 seasons in the NHL. He has spent the last eight seasons coaching the Kings, who won the Jennings Trophy for the league's best goals-against average last season.