The Hoops have not delivered the SPL and Scottish Cup Double since 2007 and, with the league title all but in the bag, they moved into the cup final on Sunday with a thrilling extra-time win over Dundee United.
Ledley said: "We haven't done the double for a long time so hopefully we can achieve that.
"It would be fantastic for us players and the staff and for the club. That's our aim and hopefully we can do it in the last game of the season.
"The cup final keeps the season alive and gives us something to play for as well.
"We've got two competitions to win - the league and, now, the Scottish Cup.
"We want to do this double badly this season and hopefully with this group of players we can do it."
Celtic will face Hibernian in the final at Hampden Park without the services of suspended duo Victor Wanyama and Beram Kayal and while Ledley admits it will be a struggle, he feels there is enough depth in the squad to overcome those losses.
He added: "It's a big blow. They are key to the way we play, especially Victor. It's going to be hard to replace them. But it gives an opportunity for someone else to come in and do well."