I recently found this picture of a building in Peking, China copyrighted in 1901. In the background is a building with a sign that reads "Young Men's Christian Association" and another sign that reads "READING WRITING AND COFFEE ROOMS". This puts it in the time period that my great grandfather was in Peking. His letter home above was written on YMCA letter head. So I am theorizing that he wrote this letter in this building.
There were at least 2 members from the YMCA that attached themselves to the Chaplain for the 14th Infantry Regiment that my great grandfather was with. So I reckon that they "rented" this building and used it as a benefit to the soldiers a long way from home. I can imagine that soldiers could read books, weeks old newspapers, magazines, etc. They were given pencil and paper to wright letters home if they wished. Top it off with coffee (maybe tea, also), which gave them a little respite from being homesick.