I'll put my copy-paste argument here:
She believes trans women aren't women. She believes trans women are only lying in order to get close to women and rape them. JK Rowling self identifies as a a TERF, or "Trans Exclusionary Radical Feminist".
Rowling donates money to the LGB alliance a UK "charity" that actively lobbies against trans rights, and has fought for the continued legality of trans conversion therapy.
Rowling has tweeted out in support of one Maya Forstater, a woman who was fired and has become a martyr for intentionally misgendering trans people, and once again perpetuating the idea that trans men are rapists. Rowling keeps pushing the idea and propping up those who agree with her.
Rowling has posted to advertise for an online shop that sells, among other things, anti-trans merch. The store is ran by an Angela C. Wild, a founding member of "Get The L Out", an organization that has in its mission statement "We stand against any kind of misogynistic politics and systems that prioritise men’s interests: queer politics and transgenderism, right-wing politics, left-wing politics" etc.
Under the pen name Robert Galbraith (a name coincidentally shared by a 20th century proponent for conversion therapy), Rowling had published multiple transphobic books, writing one book about a serial killer who dresses as a woman to lure in unsuspecting victims, and another with a self insert character who is killed after "speaking out" against transsexuality.
This is just the tip of the iceberg. There's a great video that goes into much more detail than I have here about the causes she supports and the right-wing activists and think tanks she associates with and gives money to.
I will say however, from experience, that there is basically a 1% chance any of this info would affect the opinion of somebody who supports her. But I would hope that your wife would be more willing to listen to you and consider your viewpoint.
It is admittedly difficult to argue that Rowling is transphobic. She's relatively careful with her own words and rarely does more than dog whistle. Instead you have to look at who she associates with and what causes she lends her massive finances to. And unfortunately, conservatives who are stuck in the "racism is when you say black people are bad" mindset, aren't going to be super receptive to an argument which rests on proving that JK Rowling's transphobia manifests primarily in her support of transphobic organizations and systems.