Kelly, who will stand down from his role at the NSPCC after nine years before taking on the new position in October 2010, will succeed Sir Cyril Chantler, who will have completed six years in the post.
Kelly is also chair of the Financial Ombudsman Service and the Committee on Standards in Public Life, which is carrying out an inquiry in MPs' expenses. He is a former permanent secretary at the Department of Health.
He said the fund was recognised as the leading independent commentator on healthcare in England.
"I am very much looking forward to supporting the organisation and overseeing its development at what is going to be a very testing time for the NHS," he said.
It was announced in July that Niall Dickson, chief executive of the King's Fund, would leave the organisation in January 2010 to become chief executive of the General Medical Council.