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.