。
She absolutely refused to reveal the secret.
Literal
She [topic-は] no-matter-what secret [object-を] reveal-not.
明かす ('to reveal, to disclose') is more pointed than the everyday 言う or 教える — it specifically describes letting out something previously kept hidden, often for dramatic effect. Common collocations include 秘密を明かす ('reveal a secret'), 正体を明かす ('reveal one's true identity'), 真相を明かす ('reveal the truth'). Pairing どうしても with the bare past negative 明かさなかった ('did not reveal') is enough to convey adamant refusal — no hedging like ようとしない is needed when the verb itself already carries the weight.