Nehemiah asked the king for a letter to Asaph, keeper of what, to obtain timber for gates and walls?

Answer: The king's forest

Nehemiah requested timber for the citadel gates, the city wall and his own house; the king granted it, for God's hand was on him.

“And may I have a letter to Asaph, keeper of the king’s forest, so that he will give me timber to make beams for the gates of the citadel to the temple, for the city wall, and for the house I will occupy.” And because the gracious hand of my God was upon me, the king granted my requests.”— Nehemiah 2:8 (BSB)