May has announced a fast-tracked public inquiry into the fire and said on Sunday that every household affected by the blaze would receive a minimum payment of £5,500 ($7,000) from a £5m emergency fund.
He added that teams and families of missing loved ones must prepare for "the awful reality" that some of those who died in the fire may never be identified due to the intensity of the fire. Hands cautioned that officials don't yet have exact details about the renovation that ended just a year ago.
On the lack of sprinklers in Grenfell Tower, and other buildings, he said: "My understanding is that the best expert advice is that retrofitting sprinklers may not always be the best technical way of ensuring fire safety in a building".
Anger among residents has been mounting in recent days as information about the missing has been scanty and efforts to find temporary housing have faltered.More news: Brexit talks to focus first on orderly exit
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The announcement, by Commander Stuart Cundy, who said the toll was still expected to rise, came as Queen Elizabeth II said the country was filled with sadness. "There is also a real possibility that there may be people in the building that no one knows are missing", he added. I really hope it won't, but it may increase.
Releasing a tranche of footage showing the aftermath of the blaze, Mr Cundy said: "Today, police teams continue their support to families, and make enquiries to cross check the number of those missing".
"I've never seen so many people work so hard in all my life".
Cundy, speaking outside Notting Hill Methodist Church, confirmed the previous death toll of 30 but said 28 more people likely died in the fire at the 24-story apartment building on Wednesday. He asked anyone who was in the tower and survived to contact police immediately.
Police say the harrowing search for remains had been paused Friday because of safety concerns at the blacked tower but has resumed. Cundy said emergency workers have now reached the top of the tower.
May, criticised for keeping her distance from angry residents during her visit to the charred remains of the apartment building, said on Saturday that the response to the disaster had been "not good enough".