The Mithril Mushroom Crown is located in a cave just north of Bree, where the Blacksmith is located. Once inside the cave, you will have to go through various small LEGO hatches to reach the Schematic. There are eight hatches inside the cave. The hatch that you want to enter is the one with flowers around the hatch. This hatch will take you to the key. The hatch that leads towards the lock is just Northwest of the cave. LEGO boulders are blocking this hatch, so you will have you break them to enter it. Once you enter the hatch, unlock the lock with your key to collect the schematic.
Once you've collected the Schematic, travel back to the Blacksmith in Bree. It will cost you 5 Mithril bricks, 10 Square wood pieces, 20 fish, 15 chicken pieces, and 10 pieces of bread to Forge. Once you have forged it, equip it on Thorin's head and the trophy will unlock. Also note that you will have to use the regular Thorin to meet the requirements. Any other version of Thorin will NOT count, like "Thorin (Young") for instance