。
She's been laid up sick for the past week.
Literal
She [topic-は] this 1-week sickness [due-to-で] is-bedridden.
寝込む is a compound verb 寝 ('sleep') + 込む ('go deeply into'), meaning 'to take to one's bed (with illness)' — heavier than just 寝る ('sleep'); it implies serious or prolonged illness keeping someone in bed. The で on 病気 is causal ('due to illness'). この１週間 is a time-span used adverbially — 'this past week' — without a particle, sitting bare as a temporal frame. 病気で寝込んでいる is a stock phrase for the bedridden state. Like many compound verbs with ~込む, the auxiliary adds a sense of being deep into the action — here, deep into sleep/bed because of illness.