。
She seized every opportunity to improve her English.
Literal
She [topic-は] English [genitive-の] improvement [for-のために] every opportunity [object-を] used.
The pre-noun modifier あらゆる ('every / all kinds of') is a fossil form — a relic of the classical verb 有る + ~ゆる, surviving only as a fixed determiner. It can't conjugate or take ~の; it sits frozen before nouns, paired with abstract collectives like 機会 ('opportunity'), 場合 ('situation'), 手段 ('means'). 利用する ('to use/utilize') is more deliberate than 使う; it implies tactical or strategic use — exploiting an opportunity rather than just employing a tool. The pattern ~のために is here marking the deliberate goal of the action.