You have got the axe itself right?
If so, there is a room to the north east (it is the upper most room), that has a bunch of tables in it.
The letter is on a table, drag and drop it into your inventory - this is your proof. Then, if you have the axe and the letter, you can bring them both into the room with the king. If not, you can drag and drop the letter into his chamber.
Hope that helps!