。
She took care of her younger sister, who was bedridden with a bad cold.
Literal
She [topic-は] terrible cold [from-で] is-bedridden younger-sister [of-の] care [object-を] did.
The pre-noun stretch ひどい風邪で寝込んでいる is a relative clause modifying 妹 — 'the younger sister who is bedridden with a bad cold.' Inside the clause, で marks 風邪 as the cause/reason for being bedridden — one of the productive senses of で alongside its means and location uses. 寝込む is a compound verb (寝る + 込む) meaning to be deeply asleep or, more often in this context, to be confined to bed by illness — taking the sickbed seriously, not just lying down briefly. 世話をする ('to take care of, look after') is one of the everyday light-verb expressions.