I don't think it's fair to say that J.K. Rowling is a vile person. Does she have an incorrect opinion about a scientifically proven subject? Absolutely. But to call her flat out, a vile person is just exaggeration. She wrote a whole book about the importance of love, humanity and the necessity to learn how to sacrifice certain things for a greater cause. I do believe she has a lot of humanity to her... She just does not understand, and does not want to further understand her views about gender. Does that make her any less problematic? Absolutely no.
It's like discussing with my parents subjects they are uneducated about, like sexual health, because they were not lucky enough to have this discussion or brought up in an environment where sexuality was not vilified. I would never say my parents are vile, but do I strongly hate talking about sexuality with them, because they have apparent ancient views about it. I do not have a relationship with them, but I can't say that they are bad or vile people, although they did wish me dead at one point in their lives because they were consumed with the notion of pride and ego.
Rowling needs to push past her ego, and understand that being a trans woman does not take anything away from her being a woman. I'm a gay man, and I did not understand the concept of being trans, before educating myself on the subject. I was lucky enough to not have to do that in the public light as well, and I'm sure a lot of you can say the same thing. Let's all not pretend we were all born with the notion of equality for gay, lesbian, transgender and bisexual people.
Hopefully, she'll choose to let go of her personal ego.