Another seven people at the site at the time of the fire escaped the building.
Among the dead were 13 ethnic Koreans with Chinese nationality, Yonhap news agency reported.
Kim and other police and fire officials said they could not confirm that report.
A doctor at Bestian Medical Centre in Seoul, which specialises in burn treatment, said four of the injured were in critical condition in intensive care with burns to more than 40 per cent of their bodies.
The physician, who spoke on condition of anonymity, citing patient confidentiality, said one of them was an ethic Korean from China.
China has a sizeable ethnic Korean minority and in recent years some have migrated to South Korea to seek work.
Like other foreign migrants, they often end up working in factories or doing difficult and low-paying work that affluent South Koreans have increasingly shunned.