Soon after joining Arsenal, Kennedy stepped up to the first team after Arthur Hutchins was dropped, and made his debut against Birmingham on 2 December 1922.
He impressed enough that he played all but one of the Gunners' remaining games that season, and gained international recognition, winning two caps for Ireland in their 1923 British Home Championship campaign.
However, he did not find favour under Herbert Chapman, who first switched Bob John back into defence, and then signed Horace Cope during 1925–26 as a replacement for Kennedy.
Displeased with his team's defence, Chapman had a clearout at Arsenal, and Kennedy was sold to Everton in January 1928.
After a solitary League appearance for Everton, he moved on to Tranmere Rovers that summer, playing there for two seasons until his retirement in 1930.