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Her heart was overcome with grief at the news that her older sister had died.
Literal
She [topic-は] older-sister [subject-が] died [that-という] news [target-に] sorrow [with-で] chest [subject-が] was-full.
A clausal heavyweight. 姉が死んだという知らせ uses the quotative-noun frame ~という ('the [news] that X / the [thing] called X') to embed a full clause as a noun-modifier — one of the most productive patterns in formal Japanese. 胸がいっぱい ('heart full,' literally 'chest is full') is the standard idiom for being overwhelmed by emotion; the で marks the cause of fullness. 知らせ (the noun form of 知らせる 'to inform') here refers to news or notification of an event. Layered emotion: grief on top of news on top of dying — a single sentence carrying real weight.