A lot of women prefer to avoid confrontation, especially when it comes to dating someone who they really prefer to be just a friend.
From what you say, she's always avoiding any and all advances from you so that initial suggestion that she would go out on a date with you was probably her way to let you down gently so you weren't unnecessarily hurt by a ‘No'.
Forget her as a potential dating partner and get out there and find someone else. Certainly chat to her if you like her as a friend and keep it at that level – you never know though, she may well ask you out if you no longer persue her but that's an outside chance only.
Find someone else.