。
She died after being ill for a long time.
Literal
She [topic-は] long time illness [object-を] was-doing [after-あとで] died.
病気をしていた uses する with a medical condition — 'was doing illness,' i.e., 'was ill.' あとで ('after') connects the prolonged illness to its outcome. 死んだ ('died') is the plain, direct word — no euphemism. The sentence is matter-of-fact in tone, typical of narrative accounts.