Kara would argue that a person doesn't need special powers to be a hero. Having those powers gives her a responsibility to use them wisely, but there are people who risk their lives every day to do what others refuse to. Those are the heroes worth recognizing. She's just an invulnerable girl in a cape with too many unfair advantages.
A girl in a cape who let her city down once, no matter how out of her control the circumstances had been.
"I'd ask you to prove it, but if you could do that, I wouldn't be wearing this," she plucks at the hem of her jacket, "instead of my cape. And I can hear your heartbeat. It's steady. Firm."
Might as well be honest about her (not always accurate, depending on how skilled the liar is) lie-detecting abilities if he's going to be so upfront about his secret identity. Who knew that Bruce Wayne was Batman? She certainly didn't! It makes sense, now that she's thinking about it, but the thought never crossed her mind.