Something shiny in the fireplace... Probably the key to open the drawer. But there is a fire, how to get the shiny thing? The doll is suspicious, especially when the guide to dolls have a few burnt pages.
The drawer is what opens after you solve the string puzzle, whose order is indicated by the objects on the wall. When you pull the last string it should say you hear the drawer opening, but I also included a sparkle animation in the new version to make it more obvious.
The decorations on the wall show which strings you should pull in what order. Interact with the flower pot, then the book, then the dress, then the bags (in the hall outside), then the coat. When you've interacted with them in that order the drawer will open.