。
She decided to stay with us for a while.
Literal
She [topic-は] for-a-while us [genitive-の] place [at-に] stay [thing-of-こと] [at-に] decided.
滞在する ('to stay, lodge') is more formal than the everyday 泊まる ('stay overnight') — it covers any temporary residence, including hotel stays, work assignments, or extended visits. ~ことに決める ('decide to') is one of the standard patterns for committing to an action: the action is nominalized with こと and made the target of 決める ('decide'). ところ here doesn't mean a literal 'spot' but figuratively 'one's place / one's home' — a soft, friendly way to refer to where one lives without specifying.