At least 58 people are dead or presumed dead after the blaze.
After having spoken to many members of the congregation as well as others who came in after the service to see him, Khan paid tribute to the local community for its resilience but said a feeling of anger remains. The queen was born in April but celebrates in June.
"In recent months, the country has witnessed a succession of bad tragedies", she said.
Asked if he understood the lives of people who had lived and died in Grenfell Tower, the Tory councillor, his voice faltering at times, said: "I have thought of little else. since Wednesday in the middle of the night when the phone rang, of people absolutely terrified, trapped in a high-rise building". Before that, Britons had been hit with three terrorist attacks in less than three months, in London and Manchester. "Wallowing in the wash of a general election that stripped our Prime Minister of her authority on the very eve of European Union negotiations, neither common sense nor the evidence suggest she can re-establish public confidence", wrote Mr Matthew Parris, a columnist and former Conservative lawmaker, in the Times.
The move follows widespread criticism of the local Kensington and Chelsea Council's performance as residents of the 24-storey Grenfell Tower, which was engulfed in fire early on Wednesday, condemned the relief efforts and lack of information.
Prime Minister Theresa May has been criticised for her muted response to the fire and had to be rushed away from a meeting with residents on Friday under heavy police guard as protesters shouted "Shame on you".
Queen Elizabeth and her grandson Prince William visited a community centre Friday where some survivors are being housed, with volunteers inundated with donations of clothes and food. May has announced a fund of about $6.5 million for the victims.
Reports suggest that a renovation project on Grenfell Tower a year ago intentionally did not include safety devices such as sprinklers and doors created to keep the fire from spreading.
Police have said criminal prosecutors will be pursued if there is evidence of wrongdoing.More news: OnePlus 5 promo is here: the gorgeous looks of the device
Although fire services say it could be a while before they know the exact cause of the London blaze, the horrors of what happened have been felt across the country.
The building had been home to a diverse group of residents, many of them from Sudan, Eritrea, and Syria.
The first victim has been formally identified as 23-year-old Syrian refugee Mohammed Alhajali.
'Let's get the technical advice properly evaluated by a public inquiry and then let's decide how to go forward'.
Neither side immediately released details.
"This prime minister is not viable". Residents want answers on why the fire spread so quickly, trapping numerous estimated 600 residents.
Although police did not speculate on the eventual number of fatalities, local community sources say at least 70 from Grenfell Tower are still missing, including entire families.
The tragedy has provoked a enormous response from nearby communities that have donated food and shelter to the victims.