Again, we're getting into personal preference, but the rule of thumb is that you use "'s" on every name ending in an "s" unless the person referenced is ancient (historically, not just a nonagenarian) or mythological: "Jesus' tears", "Moses' tablets", "Zeus' lightning bolts", but "Mrs. Jones's house down the street".

So it depends on, I guess, exactly which St. James it is (I assume that there must be more than one). You might also want to find out if it's his hospital or if it's just named after him. That is, whether a possessive should be used at all ("St. Mark's Hospital" vs. "St. Mark Hospital").

The real answer would be to call the hospital and find what they prefer, as it is the name of an established organization, and they might prefer "St. Jamess'sss", for all you know.