The Silver Key is an item found in the Kitchen, and it is needed to progress in the story. It is obtained by giving The Cook the Teddy Bear Limbs (which you, of course, chopped from the body of the small teddy bear), and the Cook states that he wants to show you his gratitude before handing you the Key. In Extra Mode, it is found in the Dining Room under a knocked over chair after escaping the Cook.
The Silver Key does not fit into the door in the bottom-right corner of the room (the one leading to the second floor). If the player attempts to use this key on that door, the message "The key doesn't fit at all" will appear, and a large number of cockroaches will rain from the ceiling.
The key is instead used to indirectly unlock the door by placing it in the poisoned soup on the table in the Dining Room. The key will turn black (as stated in the book on the table in the Kitchen, "The Dining of Aristocrats"), and the door to the second floor will unlock.
The other way to unlock this door is by actually tasting the soup on the table, but this results in the player's almost immediate death, and thus it does not actually work at all.