If I am not mistaken the name is now Dorothea Dix Mental Health Hospital. I remember Eastern State well as a child my brother used to torment me saying that they were going to send me there.
The last time I rode by there was in 2004 and noticed the different name.
If you contact Dorothea Dix they could let you know if they have the records and also Williamsburg has a huge genealogy group that I am sure would be willing to help you.