Kennedy suffered the attack following her daily workout, according to a statement from the Edward M. Kennedy Institute for the United States Senate.
She was the only daughter of Sen. Kennedy, a liberal stalwart who passed away in 2009. That year, Kara Kennedy accepted a Presidential Medal of Freedom from President Barack Obama on behalf of her ailing father.
In 2003, she was diagnosed with lung cancer but overcame the disease.
"Kara was a warm and caring person. Her children were the light and joy of her life," the statement from the institute said. "Her magnificent strength in her successful battle with lung cancer was a quiet inspiration to all of us and provided her family and fellow patients with hope."
Kara Kennedy was a filmmaker and television producer, according to the institute. She also served on the National Advisory Board of the National Organization on Fetal Alcohol Syndrome and as a director emerita and national trustee of the John F. Kennedy Library Foundation.
She had recently co-produced a film for the institute that was shown at its inaugural gala and groundbreaking event.
She is survived by her children Grace and Max, her mother Joan, her brothers Ted Jr. and Patrick, as well as her stepmother Victoria Reggie Kennedy and stepsiblings Curran and Caroline Raclin.
CNN's Jamie Gray contributed to this report.