A team of civil servants has been embedded in the Council office after residents complained they had been left with little support or information from officials, the BBC reported. Thirty of the victims are confirmed dead.
Police have also warned that the death toll could rise as the recovery operation continues.
In an updated press statement on Saturday evening, the council said: "Kensington Town Hall has unfortunately been closed due to yesterday's (Friday) incident", referring to the protest by residents.
He said the government is carrying out an "urgent inspection" of other tower blocks in Britain to assess safety.
King also said that there are around 4,000 tower blocks in the United Kingdom without automatic fire sprinkler protection systems in place.
The first is that, even if the regulation had passed, and required existing developers to retrofit sprinklers into older buildings, Grenfell Tower might not have gotten a sprinkler system before the fire occurred.
The response to the fire from the government and local council was "mired in confusion and hesitancy", he added - London should have acted more quickly, and Kensington and Chelsea Council should have sought help immediately.
He said the survivors would "absolutely get more than a tenner".
With blackened rubble strewn across the floor and exposed pipes, one room is unrecognisable as a home until a bath and sink come into view - the dividing wall destroyed completely by the fire.
As they left Number 10, one representative spoke to reporters briefly, saying they had spoken to the prime minister for two and a half hours about their demands and what they expected.
Residents caught up in the fire have previously condemned the relief effort as "absolute chaos".
The rules vary from place to place, as does advice about when to evacuate, but fire experts say the "shelter-in-place" directive is usually applied to buildings of 15 storeys or more.
"It is hard to escape a very sombre national mood", the queen said in a statement to mark her official birthday - an occasion that typically does not result in any public comments from the monarch.
"I saw the Prime Minister welling up", the Rev Mark O'Donoghue told Sky News.
May was criticised for avoiding locals when she visited the disaster site on Thursday and faced cries of "Shame on you" and "coward" when she returned the following day, with police breaking up scuffles.
On Friday, she was heckled on a visit to the North Kensington estate, and protesters marching on Friday and Saturday called for her resignation.
May who has promised £5 million ($6.4 million, 5.7 million euros) for emergency supplies, food and clothing, has also announced a judge-led inquiry into the disaster.
The fire at the 24-story public housing project broke out early Wednesday.