。
She'll surely keep her promise to him.
Literal
She [topic-は] surely him [with-との] promise [object-を] will-keep [probably-だろう].
Two predictability layers: きっと ('surely, certainly') sets up high speaker confidence about the prediction, and ~だろう ('will probably / surely') is the predictive auxiliary. The pair きっと~だろう is one of the most common Japanese future-prediction frames — assertive enough to commit, hedged enough to remain a prediction. 約束を守る is the standard idiom for 'keep a promise' (literally 'protect a promise'). 彼との約束 uses と ('with') + の to form a noun-attributive phrase: 'a promise with him,' the modifier coming before the noun in classic Japanese style.