As the commanding officer of the 2nd Battalion there should certainly be some information about him at the regimental museum.
Since it was reorganised and renamed in 1881, The Buffs [East Kent Regiment] has undergone several amalgamations in 1935, 1961, 1966 and 1992 such that it has now become part of The Princess of Wales's Royal Regiment (Queen's and Royal Hampshires). As a result there are several museums that are associated with the regiment.
You can follow the links to see the amalgamations starting from the regiments page of The Land Forces of Britain, the Empire and Commonwealth web site regiments.org at http://regiments.org/regiments/uk/inf/003Buffs.htm#history
On that page you will see that a in 1937 a Maj-Gen. Sir John Kennedy, GBE, CB, CMG, DSO was certainly associated with the regiment [but with missing section title I can't say in what capacity] whether or not it is the same one you seek needs to be firmly established.
A web search for - Sir John Kennedy obituary - gives more details about him in the 1st item, but not a copy of what appeared in The Times in June 1970. [note how the awards after his name have changed]
In the first instance I should contact Princess of Wales's Royal Regiment museum, click on the (Army Museums Ogilby Trust) link for email address and ask if they have any record of your John Kennedy associated with the Buffs and seek confirmation of it being the same one who became Major General.