Because of illness, the writer was unable to complete the work.
Literal
Illness [because-のために] writer [topic-は] that work [object-を] accomplishing [could-not-ことができなかった].
作家 means 'writer/author.' 成し遂げる means 'to accomplish, to achieve' — a more formal and literary verb than 終える. ことができなかった is the formal potential negative.