Is it permissible for a wife to continue working as a teacher in a school where there are no men, except for an inspector who visits his office from outside the school, given that she wears the hijab and is completely modest?

Given the situation described, the wife's work in this school is permissible, and there is no sin upon her for it as long as the husband approves of her going out. There is also no sin upon the husband for allowing her to do so. In fact, her work may be recommended and rewarded, due to the benefit it brings to Muslims.