What chance do I have of discovering ancestry files at Xining?

I am not so sure whether the authority allows you to discover that at the bureau and whether the files date back to 1856. You'd better contact them first for consultation. Here is their contact for your convenience:
email: xndaxining.gov.cn
Call: 0971-8230797

If everything goes well, you can go and hire a translator or interpreter at local travel agencies to assist you.