"The report makes clear this was a consequence of his battle with alcoholism," it said.
"We are grateful to the many friends and also medical experts who sought to help down the years but ultimately this was an illness Charles could not conquer despite all the efforts he and others made."
The family said they had been "touched beyond measure by the outpouring of warmth" from all parts of the country and all sides of politics after 55-year-old Mr Kennedy died suddenly at his home in Fort William on Monday
"The pain we feel has been at least eased somewhat by the reaction of so many people across Scotland, the UK and beyond, and by the affection expressed by politicians across the spectrum.
"It has become all too apparent to us how much Charles meant to so many people and how many lives he touched."
Mr Kennedy's ex-wife Sarah and 10-year-old son Donald attended a special session in the House of Commons on Wednesday, when MPs delivered a series of warm tributes to their popular former colleague.
"The words and images of that day, and of so many other tributes, will be there for us to look back on with pride in Charles, and Donald will always know what a special father he had," the family said.
The family thanked the public authorities and emergency services for dealing with the death "with such courtesy, kindness and efficiency" and said funeral arrangements could now be made and announced in due course.
Mr Kennedy admitted publicly that he had been receiving treatment for an alcohol problem just months after leading the party to its greatest success in the 2005 general election.
Despite declaring his hope to continue at the head of the party, he was forced to stand down in the face of the threat of resignations by senior colleagues.
He lost his Commons seat after 32 years last month as the SNP almost swept the board north of the border in the general election.
Scottish Liberal Democrat leader Willie Rennie said he hoped one legacy of Mr Kennedy's untimely death would be a better public understanding of the dangers of excessive drinking.
"We knew he had a battle with alcoholism and it was one that he ultimately lost. I can only hope that people will have a greater understanding of alcoholism as an illness," he said.
"That he was such an outstanding public figure whilst he was ill shows how remarkable he was. It is important to remember how Charles lived and not just how he died. The warmth of the appreciation for Charles' life from far and wide has touched his family and friends."