Very unusual case I have ever seen. Of course there are numerous Koreans learning English. However, I don't think there are many families that can afford 24hours working worth of salary which would be like $20,000 a month. Even if you dont count sleeping hours still too expensive for most families. Those rich Koreans they tend to do things more formal than that. Teaching with no proper visa is ok. No one really checks it and during your stay no harm would be intended as long as you would be nice to them. But don't get into money that much I can't guarantee you then. The average salary I think is $25~40 per hour depeding on how good connection you have. No student would study with you all day because it is too expensive. They would rather leave Korea to English speaking country with same budget. I am pretty sure you can get a room for free since English learners are everywhere hoping to share their house with you.