The club confirmed on Wednesday the operation, carried out six weeks ago, was "completely successful".
However, complications meant Kenwright required a "prolonged period in an intensive care unit".
"He has returned home to continue his recovery which is expected to be lengthy, but complete," the club said
In September, Everton owner Farhad Moshiri agreed to sell his 94% stake to American investment fund 777 Partners.
The club said Kenwright, 78, had "worked hard" alongside Moshiri "right up until the day of the procedure" to help facilitate the proposed takeover.
Toffees fans held protests before some home games last season and called for Moshiri and the board to leave the club.
Chief executive Denise Barrett-Baxendale, chief finance and strategy officer Grant Ingles and non-executive director Graeme Sharp have all since left their boardroom roles, but Kenwright remains in his post as chairman.
"Mr Kenwright has always been fiercely protective of his private life and is grateful that his privacy has been respected during this difficult time," Everton added.
"He hopes this will continue following this update."
