If you aren't going to use local media, it is going to require internet streaming. My recommendation would be to have the internet bandwidth increased, then have the network administrator find a way to limit the bandwidth that can be used for that specific service, such as having all the machines that would stream the music on one network, and limit the bandwidth for that network. In that case, the worst that would happen is bad lagg for only the streaming network. The rest of the hospital would be unaffected by your bandwidth utilization.
For streaming services, I really like Pandora. For personal use, it is only $3.99/month or $36/year. Pandora is $24.95/month for business. If you do use Pandora, which my oral surgen did, the patient/staff could choose what they want by creating their own station or listening to another station based on a favorite artist, and thumbs down and thumbs up songs based on what they like. You can try out Pandora for free at this site. Using Pandora for free will have commercials, but the paid version does not. There are other steaming services out there, such as Grooveshark, LastFM, Spotify, etc. I was just referencing Pandora because I actually using it and have done more research about it. I know that they offer a business version of their product and I really enjoy the paid personal version of their product.