Stretched police and social services have pleaded for help as they struggle to cope with the impact of the Calais migrant crisis.
Paul Carter, leader of Kent County Council, met officials from the Home Office to request urgent support in dealing with the arrival of hundreds of unaccompanied young migrants at the port of Dover.
It was also revealed that Kent Police have asked neighbouring forces to lend them officers to help manage the giant lorry park put in place on the M20 as a result of the chaos over the border.
The number of asylum seekers under the age of 18 in the local authority's care has almost doubled to 605 in the last three months, leaving it with a multimillion-pound funding gap.
Mr Carter said children's services in the county are dealing with a "massive logistical exercise" and warned that the surge appears set to continue.
He said: "Our social services are working all the hours that they possibly can and we have no more capacity to take many more in the coming weeks if the increase in numbers continues as in the past few weeks."
Mr Carter said the Home Office was trying to broker a solution to identify accommodation and come up with extra support for frontline social care teams dealing with the influx.
"From Theresa May down, we are trying to get some help and support to manage this crisis," he said.
"While the number of asylum-seeking youngsters continues to rise, we need to make sure we man and have the physical assets to run the reception centres, and then make more permanent placements in communities around the country."
He said the council faces a shortfall of £5.5 million in care costs.
"We get a government grant, but it is never quite enough to meet the total cost of supporting those young people," he told BBC Radio 4's World At One.
A Home Office spokeswoman confirmed local authorities are responsible for providing suitable care to unaccompanied asylum seeking children.
She added: "Together with the Department for Education, we are working closely with Kent County Council to discuss the options available to them."
Kent Police asked for neighbouring forces in the South East to provide officers to help in the policing of Operation Stack - in which long queues of lorries are parked on the M20 when cross-Channel crossings are disrupted - under "mutual aid" arrangements.
Ann Barnes, police and crime commissioner in the county, said it was the first time the step had been taken to support the operation, which she said had cost "well in excess" of £1 million in just over five weeks.
She said: "Our officers and staff - working each day to keep our strategic road network safe -- are tired and exhausted after many weeks of long shifts in difficult conditions, so we are grateful to other forces for coming to help us.
"This is the first time Kent Police has ever taken this step to help with policing Operation Stack, which just goes to show the impact this prolonged period of disruption is having on the health and well-being of staff, as well as the adverse impact on many communities.
"The fly in this ointment though is the fact that Kent Police will have to pay for these extra resources and so this will add to the ever-increasing financial burden of policing this operation."
On Wednesday there was a fresh wave of attempted incursions at the terminal.
In the migrant camp at Calais known as the Jungle, the desperation to reach the UK was exemplified by 21-year-old Ahmadi Mahmood Jan.
He said he was deported from the UK to Italy a month ago but has tried four times to get on to a train or a lorry at the freight terminal in Coquelles since making his way back to France.
"I am going to keep trying, I have to keep trying," he told the Press Association.
Earlier, David Cameron blamed the Calais crisis on a "swarm" of migrants crossing the Mediterranean.
His language drew criticism, with Labour leadership candidate Andy Burnham describing it as "disgraceful", while Ukip leader Nigel Farage accused the Prime Minister of "trying to sound tough".
The United Nations secretary-general's special representative on international migration, Peter Sutherland, said demands for economic migrants to be kept out of the UK are "a xenophobic response to the issue of free movement".
Meanwhile, 12 suspected migrants were detained after being found in the back of a lorry on the M20 this afternoon.