。
She agreed to take on a leadership role in the party.
Literal
She [topic-は] that party [genitive-の] leadership-role [object-を] take-up [nominalizer-こと] [object-を] approved.
指導的任務 ('leadership duty / role') is bookish vocabulary — 指導的 means 'leading, guiding,' 任務 means 'duty, mission.' 任務につく is a set collocation for 'take up a post or duty,' part of a family that includes 地位につく ('take a position'), 床につく ('go to bed'), 家路につく ('head home'). Note the double を: one for the role she's taking up, one for the nominalized act she's approving — the kind of stacking Japanese tolerates because the verbs are nested. 承認する is formal — used in business, government, and bureaucratic contexts. 党 in this sense usually means a political party.