。
She absolutely refused to admit her mistake.
Literal
She [topic-は] no-matter-what mistake [object-を] admitted-not.
認める (mitomeru) is the standard verb for 'to acknowledge, to recognize, to admit.' In contexts of fault or wrongdoing it carries weight — admitting a mistake means accepting responsibility, something Japanese culture values but doesn't always make easy in face-saving situations. Combining it with どうしても in the negative paints an extreme version of stubbornness: she would acknowledge nothing, despite all efforts to get her to.