French Hospital was the name used for the hospital.

I would also think that back then, that late at night, it wouldn't be uncommon for there to be a lack of staff.

Didn't he go and have dinner at the hotel with Kay before he went to see his father? So it would have had to have been late. At least it appeared that way.

And Enzo may have even decided to go there after hours as not to be seen visiting a Mafia Don.