Kaito is far from a doctor, but he can spot a lung problem as well as anyone who has been through a Compulsory Education Academy, and the best known way to treat most lung infections is antibiotics, a remedy that does not exist in the Celestial Abode. After all, their technology is more antiquated than Earth's, and doctors and healers use only medicinal plants. Of course, there are healing magicians too, like Maya, the girl who helped Kaito so much in the goblin dungeon, but their services are expensive.
Also, there are certain things that even advanced technology and magic cannot cure. They are known as the Flashing Curses.