。
She was tied up cooking all afternoon.
Literal
She [topic-は] PM continuously cooking [in-に] tied-up was.
かかりきり ('completely tied up with, devoted entirely to') is a noun derived from 掛かり ('hanging, attached') + 切り ('completely, through and through'). The adverbial 切り here amplifies into 'wholly, exhaustively.' ~に+かかりきり is the standard frame for being absorbed in one specific task to the exclusion of others. The phrasing mildly suggests the speaker views the cooking as substantial work, possibly tedious. ずっと doubles the persistence: 'continuously, all afternoon.'