。
She fell into a deep sleep.
Literal
She [topic-は] deep sleep [to-に] attached.
眠りにつく (ねむりにつく) is a literary set phrase meaning 'to fall asleep,' literally 'to attach to sleep.' More poetic and deliberate than the everyday 寝る or 眠る — it evokes the moment of drifting off as a gentle arrival into sleep. 深い眠り ('deep sleep') intensifies this into a sound, restful slumber.