I was visiting my pop in the hospital last month, and although I had my 'ukulele with me (I had just arrived in town) I did not offer to play. He had not yet heard me and I didn't want to take over the room, plus there was always someone coming in to check on something. Decided to lay low, but when he came home (the next day) I played and he loved it. I kept "forgetting" to put my 'uke away when I discovered that he would pick it up and give it a strum.
(I should add that he used to be the administrator of the hospital he was in, so I doubt anyone would have objected if I had played, but as I said there was a bit of activity, even in the evening, and we were also hoping he would get some sleep.)
I would suggest, based only on my own experience of having spent lots of time as a hospital visitor, that nursing homes and longer-term care facilities might be a better use of your 'uke power. You would have a chance to build a rapport with the audience (even if there is some turnover, there would be an opportunity to establish a relationship). Part of the fun for the patients would be looking forward to your visits.
Hospitals these days try to get patients home as quickly as possible. While patients, and even staff, might enjoy the entertainment, I don't think they are in as much need of it as patients in other facilities. Just my opinion. And of course, if you are wanting to play for someone specific, a friend or a family member, that is a different story.
Kudos to you for reaching out to people in need of 'uke joy! I'm sure you will find an eager audience.