Clara Pope said her sister was “so beautiful, so emotionally wise and intelligent and so passionate and artistic and creative and understanding”.
Dorset Police said they were “confident” they had found Miss Pope, who has not been seen for 11 days, and said the death was being treated as “unexplained”.
The discovery was made at around 3pm on Saturday by specialist search teams on land south of Swanage - close to where clothing positively identified as belonging to the teenager was found on Thursday.
Fighting back tears, Miss Pope’s sister told ITV News: “I just want to tell everybody that every minute of your hard work has been absolutely worth it.
“She is - I’m not going to say was and never will - the absolute light of my life. So beautiful, so emotionally wise and intelligent and so passionate and artistic and creative and understanding.
“And I will always, always, always be one of three. Always one of three. And I just want to thank everybody.
“Everyone. Every hug, every message has been, you know - one of three, that’s all I have to say.”
Her cousin, Marienna Pope-Weidemann, also told ITV News: “We want to thank each and every one of you for everything you’ve done.
“If there is one ray of light in this nightmare it is the compassion, humanity and community spirit that you’ve shown over the last 10 days.
“Your dedication and selflessness for a girl that many of you don’t even know has been staggering and one of the few things that kept us going.
“This afternoon the emergency services found Gaia’s body. We are absolutely devastated and unable to put those feelings of loss into words.
“We thank you for everything you have done. Our little bird has flown but will always be with us.
“The police will be releasing a statement momentarily. Please respect the family’s privacy at this devastating time and keep us in your prayers. Thank you.”
The news comes hours after hundreds of people joined searches on Saturday afternoon around Swanage looking for the teenager.
Miss Pope’s father, Richard Sutherland, was among them and personally thanked members of the public who had gathered at Durlston Country Park.
He said: “I just want to say a massive thank you for coming out. It is very heart warming and the support gives us a lot of hope.”
The searches had been organised by the Find Gaia group on Facebook and saw volunteers gather at the country park, near the main beach and to the north of the town looking for Miss Pope, who suffered from severe epilepsy.
Since she disappeared hundreds of missing person posters have been distributed across the county.
On Friday, Miss Pope’s mother, Natasha, said she had not given up hope her daughter would be found alive - saying “I believe miracles can happen”.
In a message posted on social media, Ms Pope said: “I believe in this community and I believe miracles can happen.
“I am holding onto hope, along with Gaia’s sisters and the whole family who are my strength. Please hold on with us. My eternal thanks to everyone who is out there searching for my little girl.”
Earlier, Mr Sutherland said he feared his daughter’s epilepsy may have played a part in her disappearance as she had been warned by doctors that she was at risk of sudden death from the condition.
Detective Superintendent Paul Kessell, of Dorset Police’s major crime investigation team, said: “Although the body has yet to be formally identified, we are confident that we have found Gaia.
“Her family have been informed and are being supported by specially-trained officers.
“Our thoughts remain with all of her family and friends at this very traumatic time. They have requested privacy and that we make no further media releases at this point.”
Mr Kessell said the Dorset Corner had been infirmed further forensic examinations will continue.
“This will guide the investigation in respect of the circumstances of the death which at this time remains unexplained,” he said.
“I would like to thank the community for their support and understanding while this large-scale investigation continues and their efforts in looking for Gaia.”
Detectives have arrested and released under investigation three people, a 71-year-old woman, a 19-year-old man and a 49-year-old man, named locally as Paul Elsey.
Miss Pope had not been seen since about 4pm on November 7 in Manor Gardens, Swanage.
An hour earlier she was captured on CCTV buying an ice cream inside St Michael’s Garage in the town, having been driven there by a relative.
She was said to be wearing a red checked shirt with white buttons, grey and white woven leggings and white trainers. The black jacket she had on was recovered from an address in Manor Gardens.
Forensic investigations have concluded at two properties in Manor Gardens, which are believed to be linked to the people arrested.