I found a few things about Hospital No. 11:
The hospital had 720 beds, led by G W France and was where soldiers with venereal disease were sent (WordPress.com civil war nashville)
The hospital was a former Bishop"s home ( Civil War Nashville.org)
The TN Historical Quarterly Vol 29 N0. 2 Summer 1970 PP 166-175 has an article about Federal Hospitals in Nashville, but I am unable to access it online.
I searched through several Nashville maps and only found a street called University. Not a pike. The pikes only went to adjacent towns like Lebanon, Murfreesboro, Nolensville, Granny White, Franklin, Harden (now Belle Meade), Charlotte, etc.
An University St I found was on a map online at Tennessee Virtual Archive, Map of the City of Nashville 1860. Only mention of University I found. It was located where the Western Military Institute and Univ. of Nashville stood (today is where Metro Government Bldg is located). Market St (today 2nd Ave South) and Lindsey Ave is the intersection. University St was a short street parallel to Market St. near Lebanon Pike and the Cumberland River. I don't know if it is the correct location for the Hospital No. 11.
I read an article about prostitutes in Nashville while occupied by the Yankees. Apparently to "handle" the prostitution problem, the women were forced to get a license each week if they passed a medical exam. If they failed the medical exam, they were sent to Hospital No. 11.
Hope this helps.