Police detained the person in charge of the centre, the county said. It did not identify the person or the training centre.
Most of the casualties were students aged between 7 and 16 staying on the centre's second floor, the Beijing Youth Daily said in a report, though the report was later deleted from the newspaper's website.
In a social media post, Zhonghong, a news outlet backed by China's state planner, identified the centre as Zhenxing Martial Arts Centre.
Calls to the centre and Chen went unanswered.
Fires are not uncommon in China, due partly to patchy safety protocols and sub-standard construction.
In one China's most horrific fires, 309 people were killed at a nightclub in the city of Luoyang, also in Henan, on Christmas Day in 2000. Most victims suffocated in the smoke-filled dance hall, which had only two emergency exits, and no sprinklers or smoke alarms.