Striker Kee headed home the winner in the 76th minute after Sean McConville's free-kick was headed back across goal by Ben Richards-Everton for the unmarked Kee to score.
Stanley dominated the first half with Ipswich managing just two shots to Accrington's nine.
Goalkeeper Bartosz Bialkowski denied Sam Finley early on and also dived low to keep out a strike from winger Jordan Clark.
The Polish stopper did well to prevent a Clark effort from going into the bottom of the net after 39 minutes before Ipswich had their first real shot – a Freddie Sears effort which was collected by Connor Ripley.
The Reds should have taken the lead on the stroke of half-time but Clark's free header was inches wide of the post.
After the re-start, Stanley almost gifted Town a goal with keeper Ripley scrambling back to stop a header from his own defender Richards-Everton crossing the line.
In a game of limited clear-cut chances, Middlesbrough loan stopper Ripley produced a superb save to keep out a powerfully hit strike from Jon Nolan after 70 minutes.
Stanley finally broke the deadlock, with Kee celebrating his ninth of the campaign to pile more misery on Town boss Paul Lambert, whose side sit bottom of the second tier.