She (still) might not be telling you the truth about you being the father because she may not want (you) to be the father. For that reason and for down the road legal reasons, you best do as was suggested and have a DNA match done.

As for the lady being totally in the wrong; according to you, your relationship was an on and off and a "friendship" thing. Did you expect her to stay in the closet when the relationship was not on or friendly?