The question is, “先生、もう薬を飲まなくてもいいですか?” ~ てもいいですか is in GenkiI p.152. It means “May I …?” or “Is it okay if I …?” 飲まなくてもいい is negative, so it means “Is it okay if I don’t take the medicine anymore?” The answer starts with いいえ, so the doctor is saying “No. You have to take the medicine until next week.” まで is “until,” so “いいえ、来週の月曜までは飲んでください。” is the correct answer.