。
She quickly adjusted to his way of life.
Literal
She [topic-は] soon his [genitive-の] living-way [to-に] adapted.
順応する ('to adapt, accommodate, conform') is a Sino-Japanese verb for adjusting to new circumstances — physical, social, or biological (it's also the term for ecological 'acclimatization'). The thing being adapted to takes に, the standard particle for the target of an emotional, mental, or relational predicate. 生き方 ('way of living, lifestyle') is built from 生きる ('to live') + 方 ('way, manner') — a transparent compound that productively pairs with most verbs (やり方 'how to do,' 考え方 'way of thinking'). A fairly compact sentence about settling into someone's habits.