。
She'd been ill since around summer.
Literal
She [topic-は] summer-around [from-から] was-sick.
頃 (sometimes ごろ) attached to a time noun marks an approximate time: 5時頃 ('around 5'), 子供の頃 ('when (I) was a kid, around childhood'). Here 夏頃 = 'around summer.' Combined with から ('from, since'), it gives 'since around summer' — the from-source meaning of から. The past tense だった shows that the illness extended through some period that's now past. A clean way to specify a fuzzy starting point in time.