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She doesn't reply to any letter.
Literal
She [topic-は] any letter [to even-にも] reply [object-を] doesn't-put-out.
返事を出す ('put out a reply,' i.e. send a reply) parallels 手紙を出す ('send a letter') — Japanese uses 出す ('put out') as the verb of sending mail, where English uses 'send.' The どんな~にも frame combined with the negative gives universal denial: 'to no letter at all does she reply.' This kind of broad-brush universal claim is common in characterising statements about someone's behaviour or personality.