Police detective chief inspector James Harman said Saldanha, 46, also had injuries to her wrists.
He told the inquest at Westminster Coroner's Court that two notes were found at the scene and another was found among Saldanha's belongings. He said there were no suspicious circumstances, meaning nobody else was involved in Saldanha's death.
Harman said police were examining the notes, interviewing the nurse's friends, family and colleagues and looking at emails and phone calls to establish what led to her death.
He also said detectives would be contacting police in the Australian state of New South Wales to collect "relevant evidence."
Saldanha answered the phone last week when two Australian disc jockeys called seeking information about the former Kate Middleton, who was being treated for severe morning sickness. The DJs impersonated Queen Elizabeth II and Prince Charles, and Saldanha was tricked into transferring the call to another nurse, who revealed private details about the duchess' condition.
The DJs, 2DayFM's Mel Greig and Michael Christian, apologized for the prank in emotional interviews on Australian television, saying they never expected their call would be put through. The show was taken off the air and the DJs have been suspended.
New South Wales state police said Friday that they were investigating a letter sent to the station that made several threats against the DJs. Police declined to release details of the letter.
"The safety of our employees is an absolute priority," 2DayFM's parent company Southern Cross Austereo said in a statement. "We have sensible measures in place, as we always do, to ensure our people are safe. This is now a matter for the police, and we trust they will investigate any specific threats that emerge."
The Australian Communications and Media Authority is investigating whether radio station 2DayFM breached its broadcasting license conditions and the industry code of practice.
In London, coroner Fiona Wilcox opened and adjourned Saldanha's inquest until March 26.
Wilcox expressed "my sympathies to her family and everybody who has been touched by this tragic death."
In Britain, inquests are held to determine the facts whenever someone dies unexpectedly, violently or in disputed circumstances. Inquests do not determine criminal liability or apportion blame.
The local authority, Westminster Council, said Saldanha's body was released to her family after Thursday's hearing.
Saldanha, who was born in India, lived in Bristol in southwestern England with her husband and two teenage children. Her husband, Benedict Barboza, has said she will be laid to rest in Shirva, India.
The family was not in court. Lawmaker Keith Vaz, who has spoken on their behalf, said the nurse's loved ones "need time to grieve."
Vaz said a memorial Mass would be held Saturday at London's Roman Catholic Westminster Cathedral.