The American, from Utah, was on Westminster Bridge with his wife Melissa, 46, when they were mowed down by Khalid Masood’s car.
The pair were on the final day of a trip to London to celebrate their 25th wedding anniversary.
Kurt died of his injuries and Melissa was left with a broken leg and rib and a cut head.
He was one of four people killed by Masood, who was shot dead by police.
Clint Payne, Kurt’s brother-in-law, said: “This has been a humbling and difficult experience but we have felt the love of so many people during these past several days.
“It’s been a tender experience for our family to be together with Melissa here.
“Her health is steadily improving and she has been strengthened by the presence of her family.
“She is so grateful for the outpouring of love and generosity.
“They loved it here and Kurt immediately said that he felt like he was at home, so thank you for being such good people.”
Sitting with 12 members of the couple’s family, Payne thanked the paramedics, medical personnel and those who had helped them since the attack.
He said: “So many people have been so kind and we are deeply touched by their goodness and generosity.
“Your notes, prayers, donations and love have helped us so much.
“The most difficult part of all of this is that Kurt is no longer with us and we miss him terribly.
“He was an amazing individual who loved everyone and tried to make the world a better place.”
The couple had been visiting Melissa’s parents, who were serving as missionaries in the London Church of Jesus Christ of Latter-day Saints, when they were caught up in the attack.
Her family said that Melissa’s health was improving, adding “she is so grateful for the outpouring of love and generosity”.
The deceased was described as an “amazing individual who loved everyone and who tried to make the world a better place”.
The family added: “We are deeply saddened to lose him but are grateful that the world is coming to know him and be inspired by him.”
Taking questions from reporters, the family said they were trying not to concern themselves with the attacker’s motivations and the suggestion he carried out the attack in the name of religion.
Speaking about the couple’s love of the capital, Payne said: “They loved it here and Kurt immediately said that he felt like he was at home, so thank you for being such good people.”
He added: “For those of us who knew Kurt in life, we know that Kurt would have bared no ill feelings to anyone and we can draw strength as a family from that.
“And his whole life was an example of focusing on the positive, not pretending that negative things don’t exist but not living our life in the negative and that’s what we choose to do also.”
Speaking during a press conference at New Scotland Yard, a few hundred yards from where their daughter was hurt, her parents, Dimmon and Sandra Payne, said they would travel back to the US with her when she is well enough to go home.