The hell-raising icon of stage and screen died on Saturday at London's Wellington hospital after a long illness, Kenis explained.
"He was one of a kind in the very best sense and a giant in his field," he said.
O'Toole's daughter, actress Kate O'Toole, said the family were "completely overwhelmed by the outpouring of real love and affection".
"In due course there will be a memorial filled with song and good cheer, as he would have wished," she added. "Thank you all again for your beautiful tributes -- keep them coming."
Irish President Michael D Higgins said it was "with great sadness" that he heard of O'Toole's death.
"Ireland, and the world, has lost one of the giants of film and theatre," he said in a statement.
"I was privileged to know him as a friend since 1969. I spent part of 1979 in Clifden where we met almost daily and all of us who knew him in the West will miss his warm humor and generous friendship.
"He was unsurpassed for the grace he brought to every performance on and off the stage," said the statesman.
British Prime Minister David Cameron offered his condolences to O'Toole's family and friends.
"His performance in my favorite film, 'Lawrence of Arabia,' was stunning," he said.
British actor/director Stephen Fry called the actor's death "terrible news."
"Farewell Peter O'Toole," he wrote on Twitter. "I had the honor of directing him in a scene. Monster, scholar, lover of life, genius."
O'Toole was nominated for eight Best Actor Oscars and received an honorary award in 2003.
Respected British film critic and friend Barry Norman said O'Toole "deserved at least a couple of Oscars".
"He was a very considerable movie star," he told the BBC. "He was an extremely good looking and charming guy".
The son of an Irish bookmaker, O'Toole was born in 1932 and raised in northern England.
After working briefly as a journalist and a radioman for the Royal Navy he went to study at the prestigious Royal Academy of Dramatic Art, in a class that included future stars Albert Finney, Alan Bates and Richard Harris.
After making a name in theatre, his big break in cinema arrived in the form of David Lean's 1962 desert epic "Lawrence of Arabia," in which he played British army officer T.E. Lawrence who helped lead an Arab revolt against the Ottoman Empire.
The film won seven Oscars, including Best Picture and Best Director, but O'Toole missed out on the Best Actor prize, which went to Gregory Peck in "To Kill a Mockingbird."
He was unsuccessfully nominated a further six times before being given the honorary award in 2003, which he initially refused.
His final nomination, also unsuccessful, came in 2006 for his performance in "Venus".