She never fails to write to her mother every week.
Literal
She [topic-は] every-week surely mother [to-に] letter [object-を] writes.
毎週必ず ('every week without fail') — 必ず (かならず) means 'certainly, without fail.' 母親に手紙を書く is 'to write a letter to her mother.' The emphasis is on the unfailing consistency.