The correct question should be ''Who is responsible for watching the little children...?'' because in that case 'Who' will be the most appropriate solution.
If we put Which or Whom it would be like the previous text of the question is missing and with Who this question is complete. If we leave it with Whoever is would be like the rest of question is missing.
To sigh loudly would imply that you are being snotty and you want your way, but it is more personal. You may just be sighing to one specific person instead of trying to draw attention to yourself over a bunch of people, perhaps to gain their attention.
