If you just beat the Shadow, head through the door behind where it was and then through the other door in that room. Defeat the Shadows and you'll get a Security Keycard that can be used to unlock doors like the one from before. Both use Fire and Electric attacks but are weak to Wind. Additionally, if you take on the Shadow by going through the doors behind it, you can ambush it! You need to defeat this Shadow to progress in the Palace. You can always go back and take on the Shadow at a later point if you so wish. At the top, you'll find a vent you can crawl through. To sneak by it, you need to only go to the north and climb up the green crates.

You'll now have two choices: Either go face the powerful Shadow at the top of the stairs you just passed head on or sneak around behind it.