Fifa rejected the 28-year-old’s move from Sporting Lisbon in the summer as the paperwork arrived too late.
The Portugal international has been training with Leicester throughout the winter to maintain his fitness.
His £22m deal was completed on Monday, but he is not eligible for the Foxes’ home match against Huddersfield.
Speaking about the four-month wait for the move to be approved, Silva said: “When you have a period like this, it’s very up and down.
“You have to keep a positive mentality and that’s what I’ve tried to do. I’ve kept my mind focused on my job and what I love to do.”
Silva, who was born in France, has won 20 international caps and was part of the Portugal squad that won Euro 2016.
“It’s been very frustrating, but now the difficult times are past me and I’m very happy to be close to my goal,” he told LCFC TV.
“I’ve been working hard every day to arrive at this moment and I’m in good shape.
“It’s been very important to keep training because it’s a long time not playing. There is nothing better than the game.”