I think she already had a soft spot for him when she broke into his safe.
She's a professional thief. Why did she stick around to get caught, looking at his old photos? She could have jumped out his window a minute before and he would have been none the wiser.
The fact is, he intrigued her - despite what his public image was, clearly he was not her usual mark. So she stayed a bit to try and figure him out.
Note that he did the same after she toyed with him and jumped out of his window. It's not everyday that Bruce meets someone (let alone an attractive woman!) with similar skills and who enjoyed using those skills as much as he did.
I also find it interesting that Selina admits to being the thief as soon as Bruce accuses her. She could have pulled the helpless maid act longer as she skittered around him and escaped (clearly he couldn't run after her on that cane), but she acknowledged that he was her match, being as intelligent as she was. Again, not her usual mark. I don't see her admitting to ruses that often when she's caught in one.
Although I believe that Bruce "chasing" her for the first half of the film was him trying to mentor her. Leading her onto the right path, because he felt that her abilities were being used in ways harmful to herself and others. And of course she didn't come looking for him because in her eyes, she is Miss Independent and don't need no help from no man!
But she has a crush on the Batman (who wouldn't, honestly?) and that carried over when she found out who he really was. But even before then, I maintain that he intrigued her. Even if she was in denial about it.
It's only after Bruce resolves to take the bomb over the bay (well, and Tatelia reveals herself as a murdering psychopath), that he realizes he and Selina had the same life goals all along - to be free of their pasts. And why not spend it with someone who's basically your counterpart?