Eleanor Whitelaw was taken to hospital on July 11 after being found at the property in Edinburgh's Morningside Grove.
She died at the city's Royal Infirmary yesterday evening, Police Scotland confirmed.
Officers said the incident is now being treated as murder.
Robert Buczek, 23, was charged with attempted murder and assault to severe injury and danger of life when he appeared at Edinburgh Sheriff Court yesterday in connection with the case.
Buczek made no plea or declaration and was remanded in custody following the appearance in private.
Mrs Whitelaw was left in a critical condition after being attacked more than two weeks ago.
In a statement, Police Scotland said: "The 85-year-old subsequently passed away within the Royal Infirmary of Edinburgh on Monday evening."
Confirming that the case was now a murder investigation, they added: "A 23-year-old man has made a first appearance at Edinburgh Sheriff Court and is remanded in custody."