Translation:I would stay in this hotel until next week.
I would certainly always just use "until next week".
My feeling is that with the "the", it means you're referring to a week other than the one after this one.
So: "I am in the hotel next week" (the week after this one).
But "I arrive in the hotel on the 20th and will stay until the next week" (ie the week after whatever week the 20th is in).
Personally, I would be more precise. We stay at a hotel until a certain day, not a certain week. (If someone says "até semana que vem", that leaves me as an American wondering whether the person will check out of the hotel on Sat or Sun or Mon or the following Fri or Sat or maybe even Sun.)