The next time you guys go out or hang out, sit NEXT to her, like plop down right next to her. Close enough so that while you guys talk and laugh you can make some physical contact. Bump shoulders with her to emphasize a joke or comment, touch her arm, grab her hand with an "Oh my gosh did you hear about". These are pretty innocent ways to break the touch barrier as you call it. She will reciprocate if she likes it, if she doesn't...ehhhh...could mean a lot of things. Assuming she doesn't run away from you, it should be more and more comfortable to initiate and have physical contact like holding hands, hugs, arm around the shoulder, and the eventual first kiss and whatnot.
I applaud you for taking your time
As an aside, physical contact is just another way to connect with people. It doesn't have to be sexual or whatever. It deepens the relationship. My best friend is female and we have always hugged as a greeting and as a see you next time. I wouldn't dream of any sort of romantic relationship with her, it's just what we do because we are very good, long time friends.