The letter is on a table in a room in the far north east of the second level of the dwarven mines (
map). Drag and drop it into your inventory and visit the king again (toss it into the room first, if the guards take it).
If the dwarven council member isn't near the axe monument, he very often moves into the house just south of there. If you don't see him right away, try running around the house close to the walls; if he is standing beside a wall he may not show up until you get close. One person reported finding him over in a group around the town crier.
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