A sickroom is a room where an ill person stays and rests — either a bedroom that has become one while somebody is unwell, or a room kept ready for that purpose in a school, a boarding house or a large old home. The word carries an atmosphere: quiet voices, curtains half closed, medicine on the table, a certain smell. Writers use it for that atmosphere as much as for the room itself, and you often meet it in phrases like the hush of the sickroom or sickroom duties. It is a somewhat old-fashioned and literary word today, more British than American; in modern speech people simply say "his room" or, at school, the nurse's office or sickbay. If you meet it in a nineteenth-century novel — and you will — it usually means a whole household is organised around one ill person, which is what nursing looked like before hospitals were common. Written sickroom or sick room, both fine.