Alex lied to her for years about being involved with the DEO, and she hadn't been able to tell that J'onn was really a Martian beneath his Hank Henshaw disguise. Having powers doesn't make her all-powerful, all-knowing, or all-seeing. Despite what some may think. She may have the capacity to be a lethal threat to any world with a bright yellow sun (as red kryptonite did well to prove), but that's just not who she is.
"And you're right. Danvers isn't my last name." Might as well own up to that, too. "It's Zor-El. Kara Zor-El. The Danvers' took me in after I arrived on Earth."