Kim grabbed the only goal of the first half on 24 minutes before the Ulsan, a point ahead of Pohang Steelers, quickly made the game safe after the break.
Kan Min-soo (49) and Lee Yong-joon (56) were on traget before Kim grabbed his second from the penalty spot on 59 minutes.
Pohang remain within touching distance of the leaders as Lee Myung-joo scored the only goal against Suwon Bluewings on the stroke of half-time.
Champions FC Seoul continued their revival with a third successive win that moved them up to sixth. Kim Hyung-sung netted the winner on 59 minutes to beat Gangwon FC 1-0.
Busan I'Park moved into the Championship Round places with a 2-1 home win over Chunnam Dragons.
Jose Fagner (seven) and Park Jong-woo (47) scored the goals to ensure Lim Jong-eun's 85th-minute reply was meaningless.
Seongnam Chunma missed the chance to go into the top half as Lee Ji-nam's 29th-minute goal condemned them to a 1-0 defeat at home to lowly Daegu FC.
Just one goal was needed at Gyeongnam as Milos Bosancic's 42nd-minute penalty secured victory over Incheon United.
Third-placed Jeonbuk Motors could only draw 1-1 at home to rock-bottom Daejon Citizen.
Citizen, with just one win in 19 games, led through Jung Seok-min (14) and while Leonardo (57) levelled Jeonbuk could not find a winner.