The inquest at Loughborough Town Hall heard that Ms Pilkington, her daughter and her son, Anthony, a severe dyslexic, suffered more than 10 years of abuse from a gang of teenagers living on their street.
Ms Pilkington is believed to have poured the contents of a 10-litre can of petrol over clothes in the back seat of the car, and set them alight.
The jury found that Ms Pilkington killed herself and her daughter "due to the stress and anxiety regarding her daughter's future, and ongoing anti-social behaviour".
The jury foreman said the police's response had had an impact on Ms Pilkington's decision to unlawfully kill her daughter and commit suicide.
He added: "Calls were not linked or prioritised."
The jury also highlighted lack of action by Hinckley and Bosworth Borough Council and Leicestershire County Council.
Home Secretary, Alan Johnson: 'We have to ensure there are no excuses and that lessons are learned'
Mr Johnson said the family "suffered intimidation at the hands of a local gang, culminating in a sustained level of abuse that no family should have to tolerate".
"Leicestershire Police and their local authority partners have had some hard lessons to learn about past failures, which will be the subject of further investigations," he added.
Relatives described how Ms Pilkington had complained for years about youths "taunting and abusing" her at her home, but six months before her death had told them "I give up".
Temporary Chief Constable Chris Eyre of Leicestershire Police said he was "extremely sorry" that police failed to help Ms Pilkington and her daughter.
In a statement, the Independent Police Complaints Commission (IPCC) said it would be investigating the response of Leicestershire Police to calls for assistance made by Ms Pilkington.
IPCC Commissioner Amerdeep Somal said: "This is an extremely distressing case about which there is understandably a great deal of public concern.
Ms Pilkington's family said it had been a 'terrible time'
"It appears to be a case where sustained anti-social behaviour in a neighbourhood over a period of several years has contributed to a truly horrific and tragic outcome."
She said they would be examining "how seriously the police responded to her calls for help, whether the action was appropriate and what actions they did or did not take".
Speaking after the hearing on behalf of her family, Pam and David Cassell, Fiona's parents and Frankie's grandparents, said: "This has been a terrible time for us and we wouldn't have managed without the love and support from our friends and family.
"This case has highlighted the difficulties that families with disabled children face. We know that the agencies involved have looked to see how they can improve the way they work."
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