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She insisted that I pay the bill for the meal.
Literal
She [topic-は] me [to-に] meal's bill [object-を] pay [as-ように] insisted.
言い張る is a compound verb (言う 'say' + 張る 'stretch/stick out') meaning 'to insist stubbornly,' 'to keep saying,' often with a hint that the insistence is unreasonable or unwelcome. The pattern ~ように言う ('tell to do') here takes 言い張る instead of plain 言う, making it 'insist that someone do X.' 勘定 ('the bill, the check') is the standard restaurant word for the calculated total — 食事の勘定 = the meal's tally. The choice between 勘定, 会計, and お勘定 is largely register: 勘定 sounds neutral-traditional, お勘定 is the polite version you might hear at the table, and 会計 is what's printed on the receipt or said at the cashier.