Kay had been playing with a hip problem this season but the Dons have taken the decision to operate in a bid to resolve the issue and he is set to be out for around three months while he recovers.
Flanagan has not started a league game for the Dons since May 2012 but Robinson is confident the Northern Ireland international will make the most of his opportunity.
"Antony Kay played with his injury for four weeks and he knew he'd have to have an operation," Robinson told the club's official website.
"I don't want to bring anybody in (on loan) because I've got the perfect replacement in Tom Flanagan. We're looking forward to having him back in the side and I believe he'll give the team a real lift.
"While it's hard to lose Antony, because he's been tremendous stalwart for us and he's been one of the players of the season, Tom is a top, top centre-half and I've got a really good feeling about welcoming him into the team again."
Flanagan spent the majority of last season out on loan with Barnet and, before that, Gillingham - who the Dons play at Priestfield on Saturday. The 21-year-old collected a League Two winners medal for his contribution to the Gills' title winning campaign.