The 30-year-old is expected to make his 400th appearance for the club on Saturday as they look to record their first win in five matches.
"[Britton] is an institution at the football club and there are not many players that play for so long at one club, particularly in the modern era, so it is great," Laudrup told reporters.
"400 games says a lot. Let's hope we can give him a day to remember by winning the game."
Britton rejoined the Swans in summer 2011 following a previous seven-year spell with the club.