How do I get birth records for my Institutionalized grandmother?

I found the information on my grandmother's hospital stay years ago. She was placed in the Manhattan State Hospital after the birth of my youngest aunt (May of 1920 so probably June) for depression. She lived there until she passed in August 1951 after learning of her husband's passing in May.  I have never been able to find her birth records. Giovanna Jennie (Paradiso) Discenza was likely born in Casalbore, Avellino, Italy as that's where her mother and the siblings before her and after her were born. Per her marriage to my grandfather, immigration to Ellis Island, early census records, as well as the records of her sibling's births, Jennie should have been born in Dec 1886. Unfortunately, that is the ONE year that the Ancestors Portal does not have done for Casalbore. Herein lies the problem though..the hospital censuses and the Death Index have her listed as being born in 1888. I cannot find ANY proof of that anywhere, yet it IS the same person. Would they have had a copy of her birth certificate to admit her? Would the coroner's office have had to have proof of age?  I wrote to the hospital in around 2008 but never received a reply. 

This has been so frustrating (I'm sure you all know that feeling). Does anyone have any suggestions? It doesn't help that her father is a dead end for us also. {sigh} 

Thanks in advance for any help!

Sandi