Considering the circumstances of her death, she would have been sent to Anchorage for an autopsy. If she didn't have family in Fairbanks, it is logical that she would have been buried in Anchorage. You could call the cemetery and ask what funeral home handled her burial. The funeral home would likely have records that would include the cause of death.
People freeze to death on the streets of Fairbanks every winter. But in June she would not have frozen. However, she may have died of exposure/hypothermia.
You can contact the Noel Wein Library in Fairbanks and ask for the reference desk. They have the Fairbanks Daily News-Miner on microfilm. If you give them her date of death, they will check around that date to see if there was anything in the newspaper. There is also an obitualy index at the library, but, as you say, it is unlikely there was an obituary. There is also a death index at the library, and I believe it would cover 1960.
Noel Wein Library
Telephone: 907) 459-1020
You would need only her name and date of feath to obtain a death certificate.