In an emotional statement, the devastated family also offered their thanks to those involved in the search for the 15-year-old.
A post-mortem is due to take place on Wednesday to determine Nora’s cause of death.
Her body was discovered beside a small stream, about 2.5 kilometres (1.6 miles) from the jungle resort of Dusun, on Tuesday.
Describing how the teenager had “truly touched the world”, her family said: “Nora is at the heart of our family. She is the truest, most precious girl and we love her infinitely.
“The cruelty of her being taken away is unbearable. Our hearts are broken.
“We will always love our Nora.”
The statement added: “We would like to thank all the people that have been searching for Nora and trying their best to find her.
“We thank the local people here and those far and wide for their prayers and support at this time.
“Nora has brought people together, especially from France, Ireland, Britain and Malaysia, united in their love and support for her and her family.
“To all our friends and family at home, we can’t thank you enough for all your love.”
Nora, who was born with the brain defect holoprosencephaly, disappeared on Sunday August 4 while on holiday with her family.
Hundreds of people were involved in the search and rescue operation and it was volunteer helpers who found her body.
Deputy police chief Mazlan Mansor told reporters at a press conference that the remains were winched by helicopter to a hospital mortuary.
He added that the body “was not in any clothings” and that while it remained a missing persons case police were looking into all possibilities including the “angle of criminal investigation”.
The teenager’s parents, Meabh and Sebastien Quoirin, a French-Irish couple who have lived in London for 20 years, had thanked those looking for her as fundraising pages set up by Nora’s aunt and uncle collected more than £100,000 from well-wishers.
Following the confirmation that Nora’s body had been found, Irish premier Leo Varadkar said that the situation was “every family’s worst nightmare”.
He tweeted: “Our thoughts & sincere condolences are with Nora Quoirin’s parents, siblings & wider family at this unimaginably difficult time.
“They have experienced every family’s worst nightmare. I’d like to pay tribute to everyone who searched for Nora. May she rest in peace.”
President of Ireland, Michael D Higgins, and his wife Sabina offered their “deepest condolences” to Nora’s parents, to her siblings and to her extended family.
The French Foreign Ministry also expressed its sincere condolences to Nora’s family.
The Quoirins had said Nora’s condition meant she was not independent and had difficulty walking.
Search crews looking for the teenager had played her mother’s voice in the dense Malaysian forest near where she disappeared.
Police had said Nora who was travelling on an Irish passport, was believed to have climbed out of her resort room window.
After Nora went missing, her family described how she was particularly vulnerable.
Describing her as a “very special person”, they said she would not have gone anywhere alone.