I'm wondering whether you've stopped reading, disgrace, but in case you haven't...
Many DCs are horrible to their parents, and much more horrible to their mothers than their fathers. One theory (and it rings true to me) is that some teens have to behave appallingly as part of growing up: it's their way of convincing themselves they don't need their parents - in fact that they're desperate to get away. If everything stayed lovely (the theory goes) they'd stay at home forever. The closer they feel and the stronger the bond, the harder and further they need to pull away - so the more badly they behave. Many children are closer to their mothers, and therefore have to 'work harder' at breaking that emotional bond, and do it by being foul.
A lot of parents (myself included) feel devastated by this process. You love and cherish and care for your child, and then s/he turns into a monster and breaks your heart. :(
Think about how your wife is feeling. You've told us your DD was conceived by IVF, so it seems likely your wife desperately wanted your DD and has invested an enormous amount of love in her. You DD's foul behaviour will feel soul-destroying - a total rejection of her and all the love she has given.
You are not blind. You say yourself she "went to bed crying.... she told me she felt like a skivvy... she started wailing she was a shit mum, she'd raised a monster"... If you can make the step from observation to empathy, you can understand that your DW is hurting very badly. She's full of grief and panic and confusion and feelings of failure and fear of abandonment...
You say you feel 'stupid' and you feel like you're the 'bad guy'. I think a lot of parents of teens feel like this (I know I did) because teenagers who go down this path can't be 'fixed': you try anything and everything, but in the end, the only cure is growing up. In a few years, she'll stop this awful deliberate drawn-out fighting and rejection she's busy with now, and become a decent human being again.
You can't 'fix' your DW's grief and hurt either. But you don't have to... Now, if you love her - and it sounds like you do - you just have to bear with her.
And you can give her the same gift you say she gave you: "She chose me. She made me human. She let me make mistakes, and never made me feel stupid."
So... Choose her. Make her feel human. Let her make mistakes. Don't make her feel stupid. Her feelings will pass too, as your DD grows up.