Does the Chinese Embassy in Seoul give Q1 visas to USA citizens?

We are USA citizens living and working in China with Business Visas. Our 18 year old daughter needs it to stay with us for one year (to finish high school). I think the Q1 type would be best for her. We are planning to be in Seoul for vacation this summer for one week. Would it be possible to get a Q1 (one year, multi-entry)for her while we are there (she will be with us)? If so, can you send me a link to the requirements for the Q1 and tell me how long the process is from the time the passport is submitted until it can be picked up?