The 10-year-old suffered a brain haemorrhage after she was born about eight weeks prematurely, London's High Court heard.
She is stricken by hydrocephalus, global delays in her development and learning abilities and partial left-sided paralysis.
Deputy High Court judge David Pittaway QC said her vision is impaired and she will need lifelong care and support.
Her legal team, headed by Christopher Johnston QC, claimed medics failed to advise her mother of alternatives before proceeding to a premature delivery.
He argued that, had the mother's pregnancy been allowed to continue for another two weeks, her daughter would have been born uninjured.
Sheffield Teaching Hospitals NHS Foundation Trust, however, strongly denied that its staff were in any way to blame for the girl's injuries.
The pregnancy was not a typical one and NHS counsel, Robert Seabrook QC, said medics had carried out a careful risk-benefit analysis.
He argued that, had the mother's pregnancy continued, the risk of an unplanned, emergency delivery would have been increased.
Despite its denial of liability, Mr Seabrook expressed sympathy on behalf of the trust and wished the family well for the future.
Following negotiations, Mr Johnston said that the trust had agreed to settle the girl's claim for a lump sum of £3.5 million.
And the judge had no hesitation in approving the payout, which would be used to fund the girl's care regime.
Describing the settlement as 'a good outcome', he said there was a significant risk that her claim might have failed altogether had there been a contested trial.
Addressing the girl's parents during the remote hearing, the judge said the settlement meant an end to the stressful case and offered security for their daughter.
He told the couple: “Your devotion to your daughter has been remarkable and serves as a constant example to us all.” – Strand News Service Limited.
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