The incident at the two-storey Precious Talent Top School took place a few minutes after 7am local time (04:00 GMT) on Monday.
Catherine Wambua-Soi said that rescue efforts were finished, with reports of seven deaths and 64 injured so far.
"It was a two-storey structure that was constructed using metal sheets and some wood, and a concrete slab to separate the two stories," Wambua-Soi said, reporting from the location of the accident.
"People we've talked to said the concrete slab was reinforced very poorly and the government has said it wants to investigate how safe the building was," she added.
According to Kenya Red Cross communications manager Peter Abwao, the school has some 800 pupils aged below 14.
Wambua-Soi added that the private school was located in a relatively poor neighbourhood.
"Children who come to this school would probably come from families that are not able to afford more expensive private schools, and the nearest public school was about two kilometres away," she said.
"It's schools like this one that are regularly badly constructed, made of iron sheets and concrete that is poorly reinforced."
Wambua-Soi said the mostly-poor parents might not even be able to pay for the hospital, but that the government has promised it will foot the hospital bills for those affected by this.
Wambua-Soi described the scene of the accident as one "of absolute horror", adding that children who survived came back after the rescue operation had ended to collect some of the books that were still intact, probably for reuse at a later stage.
"Unimaginable pain on the faces of the parents who have gathered here," she said, adding that the government had launched an investigation into the incident.
Many of the parents and relatives were in shock and struggling to come to terms with the tragedy, as officials tried to console them.
"I had just dropped my son to school, and heard screams on my way back, and that is when I found people assisting them out to hospital," Margaret Muthoni, whose four-year-old son was injured, told AFP news agency.
"I am just lucky my son survived with injuries. It is a very unfortunate incident because some children have died," Muthoni added.
Kepha Otieno said he lost his five-year-old daughter to the tragedy.
"I just can't believe. It is too hard for me and the family," he said.
Evanson Kamuri, chief executive of the Kenyatta National Hospital, said two of the injured children were in critical condition, while others had soft tissue injuries or light wounds and were being examined.
Local area MP John Kiarie said the area had no public land on which to construct a proper public school [Njeri Mwangi [Reuters]
Local legislator John Kiarie told KTN television that it appeared the first floor had collapsed on children on the ground floor - however, details were sketchy.
Kiarie said the area had no public land on which to construct a proper public school.
He said the disaster highlighted the lack of "regulation of educational institutions, especially those in informal settlements ... regulations that pertain to the construction and stability of educational institutions."
Moses Nyakiongora, an official with the National Building Inspectorate said at the scene: "This school was not properly constructed. It is totally substandard."
Kenya's schools have suffered several tragedies in recent years, such as a mysterious series of arson attacks targeting boarding schools that often go unsolved.
In 2017, nine teenage girls were killed in such an attack.