The blaze broke out on Thursday at the facility in the Al-Marg district of the capital, Cairo.
The head of the facility was detained as prosecutors investigated the fire.
It was not immediately clear what caused the blaze, but an initial investigation blamed an electric short-circuit, according to local media.
The fire followed a fight between two children inside the facility, officials told the Associated Press news agency.
Masrawy, an Arabic-Egyptian news portal, said the blaze was confined to one wing of the facility.
It said fire trucks and ambulances were dispatched to the scene, where the blaze was brought under control by civil-defence workers.
Egypt has seen a number of fires in recent years that have caused concerns over safety standards.
In March, at least 20 people were killed and 24 others were injured in a fire at a clothing factory in Cairo.
Last year, about 52,000 fire incidents that caused almost 200 deaths were recorded in Egypt, according to data from the country's official statistics agency.