I know that なんて and なんか are used to emphazise disgust, belittle something, say that something is undesirable and express negative feelings of the speaker. But in this context I'm not too sure if this applies...
Context: girl is talking with her boyfriend over the phone. She looks very happy and also nervous while he's telling her about his day. But her voice sounds too monotone/inexpressive, which makes her boyfriend think that she doesn't actually care about what's he saying. He tells her that she sounds like she doesn't care at all because she sounds too monotone and calm. However, it's important to say that he's not angry at all, he's just teasing her and giggling while telling her she doesn't care. Then she replies saying
But he keeps teasing her about her not listening, so she says
聞いてますよ! どうでもよくなんかないよ. While in her mind she thinks
Again, keep in mind that she says all of this without getting angry at all over his teasing or anything. She's just smiling, and in fact looks pretty happy and a bit anxious. That's why I'm not sure what's the function of なんて and なんか.
どうでもよくなんかないよ. (It's not that I don't care)
余裕なんてない = I'm not calm at all (when I talk to you)
I do notice that both sentences are in negative, so the word choice is fine, but like I said, since the girl is happy while using these words, I'm not sure if なんて and なんか are trying to emphazise or belittle 余裕 and どうでもよく or why they're being used in this context if she doesn't mean to emphazise/express disgust.