Have you ever spoken to your dad about him divorcing her? Right now, he's enabling your mother. By staying married to her and basically continuing to act like her husband only without the sex/romance, it's perpetuating your mothers delusions that he's still her husband and that his current partner is still his mistress (when it's just a legal technicality). Him supporting your mother while you and your brother were still kids was fair enough, but you've not been kids for a long time now.

You should absolutely attend your half sisters birthday. To be honest, I don't even blame her mother in this situation, I actually pity her for the situation your dad continues to put her and her daughter in. You shouldn't even feel obligated to hate her just because your mum does, she's an important person to both your dad and your sister, her being at the same family event as you should not have to be "awkward".

Had your dad divorced your mum back when all this started, and your mum got the therapy she clearly needs then she could've moved on years ago. She may have even found her own happiness with someone else by now, but she's still hung up on the idea that your dad is hers.