While playing with the youth teams of Ferencvárosi TC,[6] Kaján began her club career with the senior team in 2012. She left the team after the 2013–14 JET-SOL Liga Női NB I season, having tied for second-most goals scored on the team's season with 9.[7]
MTK Hungária FC
Kaján played for MTK Hungária FC from 2014 to 2016. In the JET-SOL Liga 2015–16 season, she scored 10 goals in 14 matches, the league's sixth-largest tally.[5] With MTK, she started at forward in 2014–15 UEFA Women's Champions League Round of 32 matches against SV Neulengbach.[8] She scored and assisted in the first leg of the 2016 JET-SOL Liga finals,[9] which MTK lost to Kaján's former team Ferencvárosi TC on aggregate.[10] She then departed for St. John's University.[5]
OL Reign
Kaján was selected eighth overall in the 2022 NWSL Draft by OL Reign of the National Women's Soccer League and signed with the team on 20 January 2022.[4] She did not appear for the Reign in her six months with the team before being transferred for an undisclosed fee to ACF Fiorentina on 27 June 2022.[2]
Fiorentina
On 18 July 2022, ACF Fiorentina announced that the club had signed Kajan to a contract through June 2023.[11]