Indian-born nurse Jacintha Saldanha, 46, was found dead on Friday, days after falling victim to a prank call by Australian radio presenters to the London hospital treating pregnant Kate.
Her tearful husband Benedict Barboza and two teenage children appeared before the media outside Britain's Houses of Parliament and spoke through lawmaker Keith Vaz, whose family is also from India.
"They just want me to say that they are extremely grateful to the public here in the United Kingdom and throughout the world who have sent them messages of condolences and support following the death of Jacintha, a loving mother and a loving wife," Vaz said.
"This is a close family. They are devastated by what has happened. They miss her every moment of every day," said Vaz, standing alongside Barboza and the couple's 14-year-old daughter Lisha.
"They will be making their own statements later on today or tomorrow to keep people informed of developments."
The family later visited the King Edward VII Hospital in central London, where Saldanha had worked for four years.
"A senior member of staff is comforting the family and offering assistance and will continue to do so for as long as needed," John Lofthouse, the hospital's chief executive, said.
"We understand that the family are distraught and need time to reflect on the assistance they require," he added. "King Edward VII's Hospital will remain here for them whenever, wherever and however they need us."
The hospital said it was launching a memorial fund to support the family and would be holding a memorial service later this week.
Vaz added that the family had met with police on Monday. Saldanha's post-mortem is due to take place on Tuesday at Westminster Mortuary in central London.
Presenters from Sydney's 2Day FM station had called the hospital at 5:30 am on December 4, impersonating Queen Elizabeth II and William's father Prince Charles in a bid to glean information on Kate's pregnancy.
There was no receptionist on duty and Saldanha answered the phone, passing it to a colleague who divulged details of Kate's severe morning sickness to the presenters.
The radio station said it tried to contact the hospital five times to discuss the prank before it aired, but the hospital denied this on Monday.
"Following the hoax call, the station did not talk to anyone in hospital senior management or anyone at the company that handles our media enquiries," a hospital spokesman said.
The 2Day FM hosts who made the call spoke on Monday of their shock at Saldanha's death.
Mel Greig and Michael Christian, who have been in hiding and undergoing counselling since Saldanha's death, said they were devastated.
British Prime Minister David Cameron described her death as "an absolute tragedy".
"I thought it was completely shocking, the news when I heard that about the suicide of this nurse, who worked incredibly hard and obviously was incredibly dedicated," he told reporters.
"I feel incredibly sorry for her and her family. It is an absolute tragedy that this has happened and I am sure everyone will want to reflect on how it was allowed to happen."