Nicola Edgington, 32, made five 999 calls on the day of the attack asking to be sectioned under the Mental Health Act because she believed herself to be a danger.
But hours later she was still at large and took a knife from a butcher's shop before she almost decapitated grandmother Sally Hodkin, 58, and attempted to murder Kerry Clark, 22.
The 2011 killing, in Bexleyheath, south-east London, came after Edgington - who now faces life in prison - was released back into the community in 2009.
An investigation by the Independent Police Complaints Commission (IPCC) found local police in Greenwich were not notified that she was living in the area after an order for her indefinite detention was lifted.
The watchdog also found Met staff failed to carry out a police national computer check on Edgington when she made contact with officers on October 10 2011, shortly before the murder.
This would have alerted them to her previous conviction for the manslaughter of her mother in 2005, the IPCC said.
Hours before she killed, Edgington sought help at a local hospital from where she called 999.
But after delays in admitting her, she walked out.
The IPCC found officers only contacted the hospital following her fourth phone call and said Edgington's second 999 call from the A&E department was downgraded because she was considered to be in a place of safety.
An officer was not dispatched at this stage despite Edgington's claim she could be very dangerous, it found.