Why does a teaching hospital have a piano (in House, M.D.)?

I don't see anything surprising in that.

There are a lot of doctors (and students, nurses and patients) who love music and appreciate its therapeutic value (or just as a way to break the tedium of some long treatment or studying sessions).

I used to study and later work in a variety of university hospitals as a doctor (before settling on a pure teaching job). All these universities have a diverse selection of musical ensembles, and classrooms are available for rehearsing in the evening hours, which is why it makes sense to put a piano there. Most places have more than one.

In fact, one of my most revered teachers used to choose his staff (besides their medical qualifications) based on whether he could use them in his chamber music ensembles, which rehearsed in the classrooms when they weren't used for teaching.

And I don't think I would have survived medical school without making music several times a week.