You do know, you need more info then that. All I know they both facing each other
If we're talking about a straight up fist fight, Batman wins hands down. If it's a battle of the minds where both agree to never engage each other in physical combat, I think Kira wins. It's always shocked me that nobody has thought that the richest man in Gotham City who owns an industrial corporation that gets military contracts might be behind the mysterious vigilante in Gotham who uses all of this high tech gear and has the financial backing to build himsel state of the art, military grade assault vehicles, boats, and airplanes. I think Light will be able to put two and two together pretty easily. Especially when you have stuff like Bruce Wayne lives with an older teen, then Batman gets an older teen sidekick. Bruce Wayne's older teen roommate moves out and he adopts a little kid, now Batman's older teen sidekick has vanished and is replaced with a little kid.
Only in comic book land are people completely unable to put two and two together on this.
Well, Ra's Al Ghul, Bane and Tim Drake (who was a kid at the time) found out Batman secret identity. So not everyone is dumb in the DC universe, only people who haven't figure out Superman secret identity is (that why you need suspension of disbelief to read comics)
Batman has fought worse. It's not like Bruce Wayne is famous in Japan either, so I don't see Kira making the connection in time before Batman kicks him in the face.
Also, since the OP didn't specify which version of Batman we're using I'm going to assume it's Adam West Batman. He beats Ryuk with anti-death god spray.