I found two web sites with articles about the TB Sanitorium: Talihina TB Sanitorium, www.undergroundozarks.com/forum/viewtopic.php?t=7166 which has a lot of pictures of the inside of buildings that aren't being used today
It was originally called the Eastern Oklahoma tuberculosis Center and in 1972 was changed to the Oklahoma State Sanatorium. On july 1, 1975 it became the Oklahoma Veterans Center at Talihana. See www.ok.gov/odva/Oklahoma_Department_of_Veterans_Affairs
You can call them 918-567-2251 and ask if they have a cemetery on site. Seems like a good possibility considering what the grounds have and are being used for.
Neither article mentioned a cemetery. Talihina does have a cemetery, Talihina Cemetery.
I couldn't find a genealogical society in that county but I did find the Leflore County Historical Society, http://leflorecountyhistoricalsocietymuseum.webs.com/http://leflorecountyhistoricalsocietymuseum.webs.com/ Hopefully they can answer your question in depth. Successful Searching!