Eileen O’Neill Burke, a former appellate judge, has won the Democratic primary for Cook County state’s attorney over Clayton Harris III, an attorney with party backing.The race is open because State’s Attorney Kim Foxx decided not to seek a third term. It was among the most spirited and competitive contests in the March 19 Illinois primary. The Associated Press called that race Friday.“It was worth the wait,” O’Neill Burke said in a statement Friday. “I am so honored to be the …