Baker, 31, enjoyed a successful trial with the club following his release from Coventry in September.
MK Dons boss Karl Robinson told reporters: "He's definitely not coming here for money. He says wants to be part of it.
"He's not one I needed, I wasn't looking for someone of that ilk. He'll add tremendous depth and strength to the group."
The winger made 182 appearances for the Sky Blues, scoring 28 goals.