Brentford boss Andy Scott told the club website: "Nicky has a proven record at this level and I am delighted we have brought him on board.
"He has played throughout the pre-season programme and will strengthen the ability of the squad."
The move was initially delayed at the start of the summer because of the managerial changes at Leicester and by Adams undergoing minor surgery.
Scott added: "I was genuinely disappointed earlier this summer when we were unable to complete the transfer but he has had no reaction to the injury.
"It was the right decision for all parties to apply some common sense and patience to review the situation when Nicky was back on the training pitch."