Kim Leadbeater, her parents and other family members viewed the ever-growing collection of flowers laid at a memorial in Birstall in tribute to Mrs Cox.
The family hugged and thanked people before speaking to the assembled crowd about the Batley and Spen MP.
Ms Leadbeater struggled to contain her emotion as she spoke of her sister’s two young children.
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She said: “For now, our family is broken but it will mend in time and we will never let Jo leave our lives.
“She will live on through Brendan, through us and through her truly wonderful children who will always know what an utterly amazing woman their mother was.
“She was a human being. She was perfect.”
Mrs Cox’s husband Brendan, who Ms Leadbeater said could not attend as he was looking after the couple’s children, sent a tweet supporting his sister-in-law.
He wrote: “Very proud of my sister-in-law Kim who spoke on behalf of us all and did her sister proud #MoreInCommon”
Ms Leadbeater said: “From a very young age, all Jo ever wanted was for everyone to be happy.
“We were brought up to see the best in everything and everyone.”
She added: “She only saw the good.”
Ms Leadbeater spoke of her sister receiving abuse on Twitter and Facebook during her political career.
She said: “But she would still see the positive and talk about the silent majority who would not always shout the loudest but were in her corner.”
She continued: “Over the last 48 hours, people have not been silent. They have been vocal and passionate and have spoken from the heart. Genuine emotion with no hidden agenda.
“Jo would have loved it.
“We have to continue this strength and solidarity for the days, months and years to come as part of Jo’s legacy.
“To focus on that which unites us and not which divides us.”
Ms Leadbeater stood with her and Mrs Cox’s parents, Gordon and Jean Leadbeater, as she paid tribute to her sister.
She spoke of having to identify her sister’s body and described the situation as “surreal”.
Ms Leadbeater said the “outpouring of genuine grief, sympathy and love” had helped the family through some “dark times”.
She thanked all those who had “expressed their love and affection” for Mrs Cox.
She also thanked the police, the two women who were with Mrs Cox on the day she died and the pensioner injured trying to help her sister.
She said: “The brave and courageous gentleman Bernard Kenny who tried to help her in her hour of need.
“Our thoughts and thanks go to him and his family and we wish him well in his recovery.”
The family spent some time at the site of the floral tributes speaking to well-wishers before leaving together.