The Wednesday blaze has so far killed 58 people while many other residents still remain unaccounted for.
"We are not interested in your reasons for not telling us sooner, we just want to understand as best as we can who may still be inside the building", he said.
The statement comes as Transport for London partially closed two Tube lines because of London Fire Brigade (LFB) saying there was a "short-term risk of some debris falling on to the tracks".
The Grenfell Tower housed about 600 people in 120 apartments.
Public anger is mounting as residents and neighbors demand answers for how the blaze early Wednesday spread so quickly and trapped so numerous tower's 600-odd residents. Hands cautioned that officials don't yet have exact details about the renovation that ended just past year.
Mr Cundy said it was already clear that the total number of the dead "will be higher" than the figure previously given by police but said he would not "speculate" on a figure.
Commander Stuart Cundy The blackened remains of the lift in Grenfell Tower.
He said five of the dead had been formally identified, and it would be a slow and painstaking task to identify the others because of the intensity of the fire.
"Where offences have been committed, I will do everything in my gift to make sure they are brought to justice", he said. Survivors of the building claim that cheap materials for the cladding and a lack of maintenance on the building were to blame for the fatal fire.
Cundy says the new number may change as investigation continues.
He says tower residents who survived fear a cover-up will keep the truth from coming out.
He said it had been "incredibly emotional working in there".
British Prime Minister Theresa May, criticized shortly after the blaze for failing to meet with victims, says the public inquiry into the tragedy will report directly to her.
He said the community in north Kensington, west London, was "frustrated and, yes, angry".
Those who lost their homes would be rehoused within three weeks, she said.
Cundy explained that one of the reasons identification has been so hard is because dental records are needed from victims who hailed from different countries around the world. This relies on dental records, fingerprints and DNA when possible and also features like tattoos or scars. Five people who died in the blaze have been identified and 74 were missing and presumed dead following the devastating blaze in the council-owned Grenfell tower block in west London.
The 91-year-old monarch said it is "difficult to escape a very somber mood" on what is normally a day of celebration.