The local authority is without a leader after a series of failed votes.
Reform UK is the largest party there following the 7 May elections, with 29 out of the 69 seats, but falls short of a majority.
The Greens have 12, Independents 11, Conservatives nine, Liberal Democrats five and Valley Independent Group three.
At a recent meeting, Reform group leader Councillor Sarah Wood and Green Party head Councillor Andrew Cooper nominated themselves for the next council leader role.
However neither was able to secure a majority after two votes, and the chamber descended into debate.
During Thursday’s meeting Brabin said: “I know we’re all keen to continue working on a cross-party basis for the good of our region, and it’s something I hope, you know, I’ve done consistently since I’ve been mayor.
“Of course, there will be times that we do disagree, but it’s part and parcel of politics.”
She added: “We’ve all been elected by the people of the region to deliver improvements.”
Kirklees Council is set to reconvene for a third time on 15 July to select a leader, according to the Local Democracy Reporting Service.
