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She absolutely refuses to agree to a divorce.
Literal
She [topic-は] no-matter-what divorce [to-に] agree [refuses to-ようとしない].
Two key pieces lock the meaning. どうしても ('no matter what, by any means') as an adverb intensifies refusal — when paired with a negative predicate, it conveys 'absolutely won't.' ~ようとしない is the negated form of ~ようとする ('try to'), expressing active refusal or stubborn unwillingness rather than mere inaction: 'won't even try to.' Note 同意する takes に for the thing being agreed to, not を — a small syntactic detail worth remembering. The sentence is a portrait of dug-in refusal.