。
She insists, to the bitter end, that she's right.
Literal
She [topic-は] herself [subject-が] right [quotative-と] saying [to-the-end-あくまで] asserts.
あくまで (kanji 飽くまで, literally 'until one tires of it') is an adverb meaning 'to the bitter end, persistently, stubbornly' — paints the insistence as relentless rather than reasoned. 主張する ('assert one's claim, push one's argument') paired with あくまで is a classic collocation for digging in one's heels. Note the chained と-quotation: 正しいといって ('saying that she's right') feeds into the main verb 主張する, so the と here is the quotative complement of 言う, not of 主張する directly.