Gudjonsson has secured his release from Spanish side Real Betis on a free transfer and has been destined for an English club.
He spent last season on loan at Wolves, after a stint at Aston Villa the previous campaign, and had been keen on a return to Molineux.
Earlier this week, the midfielder trained with Leeds United but he has now signed a two-year contract with Leicester.
Foxes boss Micky Adams feels he has snared a bargain in persuading Gudjonsson to move to The Walkers Stadium.
"Joey is a strong midfield player who we hope will win the ball and play it simple in the middle of the park," Adams told the club's official website.
"Real Betis paid £3 million for him in 2001 and he has spent time on loan at both Aston Villa and Wolverhampton Wanderers."
Adams has also struck a deal for experienced defender Makin, who is available on a free transfer after quitting Ipswich Town.
Makin has been training with The Foxes for the past few weeks and has concurred on a 12-month deal which will commence on August 1.
The Foxes chief added: "Chris has impressed during the two trips and he will sign for us on a one-year contract from the beginning of next month."
