Kai smiles when Yu is at the Palace talking to Rin who looks like he's backed into a corner. It doesn't look like she came here for games, he's not surprised she figured it out. She is a great detective after all.
He steps in closer and overhears Rin spilling everything they know about Grimm Lyra, it's basically the exact same thing Bakero told her, the difference is that Rin knows where the evidence is.
Kai's eyes widen when he overhears the name of the person. He races past them and starts investigating the person. He finds nothing but goes back downstairs with a photo in hand asking the Palace staff what they know.
Kota mentions that he is a Royal Guard at the Empress's Palace. He's met him before at a gay club consoling his friend who just got dumped.
"I even have his number."