How do I solve (mithril)?

You find the mithrill on the mountain board after leaving Mirkwood. You have to play as Farin to find it. It is in one of the hidden wall secrets, I believe the only one on the board.

As far as I know, you can't complete this side quest as any other charachter.

You don't have to go through Mirkwood to find the Mithril. The mithril is in the Mount Gundabad level. Towards the beginning of the level the path will split for a short distance only to converge again. If you take the upper path on your right you'll find a small cave. Using Farin you'll find a breakable wall. The mithril is inside a chest on the other side of this wall. Make sure you have the quest accepted when you open the chest. Otherwise you won't find any mithril inside it. This is the only way you can acquire it.