。
She finally summoned up the courage to ask him for a loan.
Literal
She [topic-は] finally courage [object-を] [working-up-出して] him [genitive-の] loan [genitive-の] application [object-を] did.
勇気を出す ('summon up courage') is a body-action idiom — literally 'put out one's courage,' as if courage were something you could pull out and offer to the moment. The te-form 出して chains it as the manner-step before the main action. 借金の申し込みをする stacks two の-genitives ('the application of a loan') wrapped around the noun-phrase backbone, with をする making it a verb-style action. Asking for money in Japanese culture is socially fraught — the long lead of ついに ('finally, at long last') and 勇気を出して ('working up the courage') hints at how loaded such requests can be among friends and family.