Residents of the tower said Mrs May was far too slow to visit the stricken community, that the 24-storey building had been unsafe and that officials have failed to give enough information and support to those who have lost relatives and their homes.
"The scale of this tragedy is clearly proving too much for the local authority to cope with on their own", Mr Khan said in an open letter to Ms May.
He said: "The conditions due to the fire damage verge on indescribable, which is why this will be such a lengthy operation taking weeks to complete".
The Metropolitan police have officially begun a criminal investigation into the fire.
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".
The first victim to be identified was Mohammed Alhajali, a 23-year-old Syrian refugee.
Following criticisms of the Kensington and Chelsea Council's handling of the disaster, Paget-Brown said "lessons must be learned".
On the night of the blaze, Choucair had not been at home, but rushed back to find the building alight after a friend inside had called, begging for help. "So that 58 would include that 30", the spokesman said Saturday.
"This is an absolutely very bad fire that took place".
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Speaking outside after the service, which he attended with his wife Saadiya, he said people are "angry not simply at the poor response in the days afterwards from the council and the Government, but the years of neglect from the council and successive governments".
"The prime minister is distraught about what has happened", said Green, who was appointed May's deputy in the wake of the general election.
He said the government is carrying out an "urgent inspection" of other tower blocks in Britain to assess safety.
The move came as the official response to the crisis drew fresh condemnation from residents and London Mayor Sadiq Khan.
More than 20 police officers rushed into the town hall to restore order and mounted police were on stand-by.
The government has committed 5 million pounds for clothes, food and emergency supplies for the victims.
It tore through all floors of the building and took more than 200 firefighters and 24 hours to bring it under control.