。
She has prior engagements all next week.
Literal
She [topic-は] next-week [contrast-は] all-along prior-engagements [subject-が] are-packed.
先約がつまっている — 先約 (せんやく, 'prior engagement') is literally 'previous appointment,' and つまっている (from 詰まる, 'to be packed/stuffed') describes a schedule being full. ずっと ('throughout') emphasizes the entire week. The は after 来週 creates a contrastive topic: 'as for next week specifically.'