The incident took place overnight Monday, the Taiwanese Central News Agency reported.
Ten of those injured were in critical condition while six others had sustained minor wounds, the outlet added.
Firefighters received reports about the fire at 04:36 a.m. on Monday (20:36 GMT on Sunday). Seventy-nine units including a total of 243 firefighters were reportedly dispatched to the site of the incident. They extinguished the fire by 05:27 a.m.
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Firefighters and rescuers have so far managed to evacuate 31 patients of the hospital.