[Breathing slowly has been successfull up to now. Though Mikami still seems somehow euphoric, he now speaks clearly. These are things he has thought about millions of times, even though ultimately it didn't matter. The 'how' was unimportant compared to what one would do with this power.]

Back when Kira had just arrived he couldn't kill L immediatly. That means that the power to kill can not affect people who Kira knows nothing about. There needs to be something Kira can identify them with. The likeliest thing would be their appearance, I guess.
In addition the powers aren't limited to one person. I'm not sure of how to get them, but it seems to be possible as once a secound person with the ability to kill by heart attacks has shown up.