The building was already blighted at the time but, 13 years of blighted mess is too long. There is no incentive for them to do anything because they pay no property taxes on the property.
The foundation also owns the properties at 3425 and 3535 Bienville. The first course of action is to call 311 and request that blight fines be levied on the St. Margaret's Daughters property on the corner of Jeff Davis Pkwy and Bienville.
311 gets the process in play.
Follow up with calls, or better yet, visits to Code Enforcement, your Council member, and the Mayor.
And, if angry enough, you could follow up with your State Representative and State Senator and inquire as to why this non-profit is being allowed to let this property languish. You can Google St. Margaret's Daughters for more info but their office location is listed as 3419 St. Claude if you want to stop by to visit.
https://health.usnews.com/best-nursing-homes/area/la/st-margarets-daughters-home-195437