。
She absolutely refused to let him into the apartment.
Literal
She [topic-は] no-matter-what him [object-を] apartment [into-に] [allow-to-be-let-in-入れさせ] did-not.
入れさせる stacks the causative ~させる on top of 入れる ('let in'), giving a permissive 'allow [him] to be admitted.' The construction is slightly redundant, since 入れる already encodes 'let in' — the causative layer adds an extra degree of permission. Negation plus どうしても turns this into uncompromising refusal: she simply would not allow it under any circumstances. The choice of アパート (loanword) signals a modern urban setting; in Japan, an アパート is typically a smaller, older, often wooden multi-unit building distinct from the more expensive concrete マンション.