。
She poured all her effort into writing the letter.
Literal
She [topic-は] with-all-one's-might letter [object-を] wrote.
一生懸命 here reads as 'with full focus and care,' not 'fast' — fitting for letter writing, which in Japanese culture is treated as a deliberate, almost ceremonial act. A handwritten letter (especially to elders, supervisors, or in formal apologies) is composed slowly with attention to brushstroke balance and word choice. 手紙 ('letter') uses 紙 ('paper') as its second character, embedding the physical artifact in the word itself.