The 23-year-old, who came through the ranks at Plymouth Argyle, spent only one full season at Kenilworth Road, but in that time he scored 18 goals in all competitions.
Vassell, a cousin of former Aston Villa and England striker Darius Vassell, told the official Blues site: "I am absolutely buzzing, I can't wait to get going.
"It's been going on for the past couple of months, that sort of thing, and once the club put in a bid that was accepted, I was just so excited and I am looking forward to moving on to the next chapter in my career now.
"It's a massive club, I am delighted to be here. And to work under Harry Redknapp, such an experienced head, I can't wait. I want to do the best I can. From what I have heard from everyone the club is moving forward and it's going to be a good place to be at."