Theresa May has said the support for families "who needed help or basic information in the initial hours" after the Grenfell Tower disaster "was not good enough".
The Prime Minister said there had been "huge frustrations" on the ground as people struggled to find information.
After a two-and-a-half hour meeting in Downing Street with victims, residents, community leaders and volunteers, Mrs May said she had heard their concerns.
She added: "The response of the emergency services, NHS and the community has been heroic.
"But, frankly, the support on the ground for families who needed help or basic information in the initial hours after this appalling disaster was not good enough."
Mrs May said money from the £5 million emergency fund to help victims buy food and clothes was already being handed out and more funding would be allocated if needed.
She has ordered daily progress reports on housing for those affected.
Mrs May added: "There have been huge frustrations that people do not know who to talk to, that they can't get through on the council hotlines.
"I have ordered that more staff be deployed across the area, wearing high visibility clothing, so they can easily be found, dispense advice and ensure the right support is provided. Phone lines will have more staff.
"Victims have concerns their voice will not be heard, that their many questions about this tragedy will not be answered.
"That is why I ordered a public inquiry, with the costs for providing victims with legal representation met by Government.
"The inquiry will be open and transparent. Government and ministers will cooperate fully. I anticipate the name of the judge will be announced within the next few days and that an open meeting will be held with residents to help shape the terms of reference.
"It has been decided today that the public inquiry will report back to me personally. As Prime Minister, I will be responsible for implementing its findings."
Mrs May said councils have been told to complete urgent safety checks on all high rise buildings further action "will be taken" if needed.
"After something this traumatic, many people will be in need of not just financial support, but also counselling. NHS London will provide that support," she added.
"I can also announce that NHS London will provide specialist long-term bereavement support for the families who have lost loved ones, and immediate psychological support is being provided by Cruise and Red Cross."
Mrs May said the Grenfell Tower blaze "was an unimaginable tragedy for the community, and for our country".
She added: "My Government will do whatever it takes to help those affected, get justice and keep our people safe."
Sir Ed Davey, the Liberal Democrats' home affairs spokesman, said: "Theresa May is acting like a hopeless by-stander rather than Prime Minister.
"We are several days on, and her belated act of contrition is not enough.
"Despite the heroic work of the emergency services, local people report that the support for survivors has been a shambles.
"This is a national disaster – yet the Prime Minister has only called a 'mini Cobra' meeting and has largely left the response to the council and charities. It is not good enough."