。
Won't you come stay with us for a week next year?
Literal
Next-year one-week stay [to-に] honorifically-come-not-[question-か].
いらっしゃる is the honorific form of 来る/行く/いる, used to elevate the listener's coming/going/being. いらっしゃいませんか is the negative-question form used to make a polite invitation — 'won't you...?' The ~ませんか structure softens the invite into a question, more inviting than a direct request.