Kjemhus started his career in minnows Bønes IL before joining the ranks of
SK Brann. After a successful time with their junior team, Kjemhus started his senior career in
Fana IL in 2014. After one season there he signed with Sogndal. He made his Norwegian Premier League debut in March 2016 against Bodø/Glimt. February 5, 2019, Kjemhus signed for
Nest-Sotra. However, after not playing once, he featured for the
Norwegian School of Economics students' team
NHH FK in the autumn of 2019. ==Career statistics==