An Australian DJ who made a hoax call to a hospital caring for the Duchess of Cambridge made a tearful apology as a coroner ruled the nurse who answered the call took her own life after it was broadcast.

Jacintha Saldanha, 46, a night sister at King Edward VII hospital in London, where the pregnant duchess was being treated for severe morning sickness, was found hanged three days after the prank call by Melanie Greig and fellow DJ Mike Christian pretending to be the Queen and Prince Charles.

The inquest heard Saldanha blamed herself for transferring the call, which she believed was genuine, on to the nurse caring for the Duchess who revealed details of her condition, broadcast on Sydney's 2Day FM radio station on 4 December 2012.

Recording a verdict of suicide, Westminster coroner Dr Fiona Wilcox said she was satisfied Indian-born Saldanha, a mother of two, had killed herself.

In an unusual move, she consented to Greig, who attended the inquest but was not permitted to give evidence, making a tearful personal statement at the end of the hearing.

As Saldanha's husband, Ben Barboza, son Junal, 18, and daughter Lisha, 16, sat motionless, Greig sobbed: "I'm so sorry for your loss. I have wanted to say that for so long."

The presenter, who no longer works for 2Day FM, added the tragedy would stay with her as "a constant reminder", and said she wished she had "tried harder to stop the prank from airing".

"To fellow announcers and DJs I urge you to speak up if you don't feel comfortable and to consider the feeling of others when trying to make a joke. The joke should always be on us, the DJs‚" she said.

After the verdict the family flanked Labour MP Keith Vaz, their spokesman, as he condemned the hoax, which was "the cause of such mirth in Australia"‚ but had "deprived Ben of his wife, and Lisha and Junal of their beloved mother".

"These despicable and cruel actions, this hoax, has changed their lives forever," he said.

The family accepted the verdict, he said, and was taking advice over possible future legal action.

The family thanked all those who had supported them over the 21 months since her death, including the Duke and Duchess of Cambridge‚ "who publicly and privately have supported the family and always been of concern for their welfare".

Saldanha's son, Junal, added: "Dad, my sister and myself miss our beloved mum every day, and will do so for the rest of our lives."

Southern Cross Austereo has donated $500,000 to a trust fund for Jacintha Saldanha's family to help them rebuild their lives.

"We do not assume, of course, that this donation or any amount of money could relieve the feelings of loss felt by Ms Saldanha's family. But it is our hope that it may help them," a spokesperson said in a statement.

The radio station conveyed its condolences but did not accept any blame for Ms Saldanha's death.

During the inquest the family's barrister, John Cooper QC, had raised questions about support offered by the hospital to Saldanha after the hoax, and whether it was appropriate that nurses also fielded telephone calls to the hospital during the night.

The coroner concluded the support offered had been "reasonable and appropriate" and that Saldanha's suicide was "not reasonably foreseeable".

Saldanha had sent several emails to colleagues describing her distress and blaming herself for transferring the call. One, sent hours before her death to a night nurse, said: "I don't know how to face the bosses tomorrow. I feel so ashamed of myself."

Giving evidence, John Lofthouse, then the hospital's chief executive, said hospital protocol dictated Saldanha should have taken the caller's number and ended the call so identity could be verified. Some senior managers thought both nurses should be disciplined, but his view, and that of the matron, was that the nurses were victims and "categorically" they would not be disciplined.

Staff believed her to be "a well balanced, robust, stable individual" and "no alarm bells rang" over her behaviour after the hoax. If anyone had "any inkling of any fragility" or thought it possible she might take her life, he said: "We would have put our arms around her, literally and metaphorically."

The coroner determined Jacintha took her life on Thursday 6 December – two days after the hoax and the day after it received global media attention. A police search of her laptop showed she researched suicide prevention sites, and news reports of the hoax.

She left three notes. In one to hospital management, she wrote: "Please accept my apologies. I am truly sorry. Thank you for all your support." Another was addressed to a junior nurse who had accused Saldanha of "bullying and harassment", a grievance that was not upheld. The third indicated "personal wishes".

Four calls – the longest lasting 45 seconds – were made by the radio station to the hospital after the hoax call to gain consent to broadcast. They were terminated by the recipient, who was almost certainly Saldanha. The coroner said: "If she did take those calls I find it inconceivable she would have consented, as a participant in the call, to its broadcast."

She concluded: "I am satisfied that Jacintha Saldanha took her own life. At the time, the hoax was clearly pressing on her mind as were the difficulties she had been experiencing with her colleague."