I don't know, if Kyouko really love and care about Kurumi like she said, she wouldn't have wish for her to break up.
She's wishing for them to break up because she loves and cares about Kurumi. She thinks Nika is a negative influence on Kurumi's life and mental well being. Is she correct? I wouldn't say so, but she's well intentioned.
One could argue that it's the reverse: she thinks so little of Nika partly (or mainly) because she wants them to break up. She basically had a single negative encounter with Nika and has refused to change her opinion of her since then despite all the evidence that she's very serious about her relationship with Kurumi. Kurumi has only positive things to say about her, and she even showed Kyouko the scarf Nika knitted for her. If that isn't proof that she's not some slut/cad just playing with Kurumi's feelings, I don't know what is.
So I think there's some evidence that at this point, Kyouko hates Nika simply out of jealousy. She says she wants them to break up out of concern for Kurumi, but there has to be an element of selfishness to that as well, whether she admits it or not.
To be fair, everyone in this story has acted selfish at some point. I agree with you that Kyouko has refused to change her opinion of Nikaidou, but we also need to consider that Kyouko hasn't seen the more caring side of Nika like we have (and even among us readers, there's been a good deal of debate about how serious Nika's love for Kurumi is).
What Kyouko knows about Nikaidou is: 1) the things she's seen: the Karaoke night, and also Nika not going home with Kurumi because she was meeting Jun; 2) the rumors, which are mostly true anyway; and 3) the things Kurumi says about her, but love is blind and Kurumi's a bit of a dolt, so that doesn't sound much credible. Also, she seems to know that Nikaidou was dating Kurumi while being in love with someone else, and that does paint her in a bad light.
I think it's plausible that, in Kyouko's eyes, Nikaidou still looks like a bad person, someone who's going to hurt her dear friend someday.