At the top of the page, click "Search Museums" to get to the Directory. Includes info on the governing authority and attendance figures.
Finding Financial Information
Many museums are private, tax-exempt organizations and as such, they file an IRS form 990, which is publically available. They do not have to make their annual reports (which often contain information such as donor lists, attendance figures, etc.) public, nor must they make the sources of their funding public.
A guide to the different parts of the 990 and what they mean created by the Foundation Center.
Finding Information on Other Funding Sources:
Special funding for specific exhibitions is credited on introductory wall texts and in exhibition catalogues, although those donors may or may not have provided funding for the entire exhibition.
Other sources of funding may be a bit more tricky to uncover. Members of a Museum's Board of Trustees make annual donations but the amounts vary widely from institution to institution. Members' publications often list donors under specific amounts, but you will not find these publications in a library (other than the museum's own library). Newspapers will cover major financial events (such as the financial rescue of MOCA a few years back). The library's LA Times database will be your best source for local museum news.