Kielsen was originally a mariner, and then a police officer with Rigspolitiet from 1996 to 2003 in Upernavik and Paamiut.

Kielsen entered politics in 2005 when he was elected to the Greenlandic parliament for Siumut and to the municipal government of Paamiut.

In the government of Aleqa Hammond, he was the minister for housing, nature and the environment.

He became acting premier in October 2014 when Hammond stepped down and the Siumut party subsequently elected Kielsen as its leader,[1][2][3][4][5][6] awarding him 44 of the 65 votes.

"[11][12] On November 29, 2020, Kielsen lost reelection to the chairmanship of the Siumut Party to Erik Jensen.