How do I reassemble the key?

First, find the key (point B, up the ladder).

You also need a ladle (in the kitchen, point H).

Now, head to the blacksmith (point C) and get a couple of items there (forge
hammer and hacksaw).

In the adjacent room (point D) there is some charcoal you need.

Now, put that charcoal in the brazier in the blacksmith room.

Make a fire and set the charcoal ablaze.

You found two items at the beginning of the game which will make a fire.

Use the flint on the sulfur to start a small blaze.

Go over to the bellows, and use them.

Now you can start mending the key.