how did takada finde mellos name?

This is very wrong. toi need a real picture to see the name of the person with the Shinigami eyes, which she had. Once she was kidnapped, she was looking at his face when he took his mask off and saw his name. I'll admit that I don't know how she got the Shinigami eyes since Ryuk was never around her but she did have them. Plot Hole I'll assume. Don't say things toi don't even know. If light knew Mello's real name this whole time, he would of killed Mello a long time ago. Again, there's some plot holes with that because Light was clearly listening to his father when he a dit Mello's real name out loud.