娘はいつも、約束を守らない母親に苛立っていた。

、。

The daughter was always irritated with her mother who wouldn't keep her promises.

Literal

Daughter [topic-は] always, promise [object-を] keep-not mother [target-に] irritated-was.

The relative clause 約束を守らない modifies 母親 ('mother'). 苛立つ ('to be irritated, fret') is a verb form of 苛立ち. 母親に苛立つ ('be irritated at/with the mother') uses に to mark the target of the emotion. ~ていた (past continuous/habitual) frames this as an ongoing state over time.