。
She insisted I should pay the bill.
Literal
She [topic-は] I [subject-が] bill [object-を] pay should [quotative-と] insisted.
主張する ('to assert, to claim, to insist') is a slightly formal Sino-Japanese verb often used in arguments, debates, or written analyses. The embedded clause uses ~べきだ ('should, ought to') for moral or social obligation — a strong recommendation rather than a mild suggestion. 勘定 ('bill, check, accounting') is a slightly older-feeling word for the restaurant bill — many speakers today use 会計 ('accounting / settling') or simply お会計 instead. The traditional way to ask for the bill in a restaurant is お会計お願いします.