You're definitely on the right track. What you're looking for is in Cordelia's room. It's not the birdcage, but somewhere more toward the left edge of the screen. The exact thing you're looking for is in spoiler text:
The necklace on the left side of room
If you haven't tried using your powers, you might want to try that in a certain place in the apartment to help clue you in here. That should definitely clue you in for what you need to do here.
You are trying to distract someone, but not Keith. You're trying to distract:
McAdams
If you need a full spoiler of the entire section of this puzzle:
You have the tracker pen and also hopefully Cordelia's necklace in your inventory. Combine them so that the tracker is tracking the necklace. Use precognition powers in the living room to learn that the kitchen table will be searched. Give Keith the necklace and put the tracker pen on the table McAdams will search. This will distract McAdams and solve that section of the puzzle
Good luck!