Kjærgaard got his debut at the age of 17, on 8 December 2015 against AB in the Danish Cup, that Lyngby won 3-0.
[6] He played his first game for the club in a 4-1 victory on 28 April 2018 against Sandefjord, where he entered the pitch in the 76th minute.
On 5 March 2020 the club confirmed, that Kjærgaard had signed a contract until the end of the season.
On 7 February 2023 it was confirmed, that Kjærgaard had returned to Norway, signing a deal until the end of 2025 with Eliteserien club HamKam, which had Danish Jakob Michelsen as head coach.
[11] On August 5, 2024 it was confirmed that Kjærgaard returned to Denmark when he joined Danish 1st Division club AC Horsens on a contract until June 2026.