She is used to staying up late at night.

Literal

She [topic-は] night late [until-まで] being-awake [to-のに] is-accustomed.

夜遅くまで起きているのに慣れている — another 'used to doing X' pattern with のに慣れている. 起きている ('being awake') is the ongoing-state form of 起きる.