If she likes me why does she stay so distant?

First, you can't just listen to her friends. She may be too shy to tell you her feelings, but you don't know for sure if her friends aren't sh*t-disturbing the situation or trying to put her on the market for a get-better rebound guy, if you know what I mean. After a relationship ends, a girl is very vulnerable. She can even convince herself that she likes someone else if it makes her less alone or feel better so that she can move on. Not to say that this girl doesn't like you, but one day you should just stop her and get her to hang out. Sometimes a shy person needs someone upfront to start a conersation. Try saying;

"Hey, we never get to chill. You're always so busy! Y'think that maybe you can squeeze me into your schedule sometime?"

Make it sound just friendly, in case you are shy yourself or so you don't scare her off from dating you if she isn't ready. If you're that unsure, invite her AND her friends for a fun day out since they all seem to like you. Having her girls around may make her feel a lot less on-the-spot. Go for it, but carefully. I think you can do this!