If I were Kira, I'd obviously want to get L's real name. The Head Doctor hasn't made a secret of his, and knowing it wouldn't get Kira very far if he couldn't be certain what the exact repercussions of his murder would be. At this point in time, proceeding with caution would be the best course, but with an enemy like L around to potentially ruin whatever plans Kira might come up with, not to mention the problems he poses back home and how ultimately knowing L's real name could benefit Kira, that's the number one name he'd want and the number one person he'd want eliminated.