Great Ormond Street Hospital's museum covers the history and personalities connected with the hospital since its inception in 1852. It also houses editions of Peter Pan from all over the world.
Don't expect to get a good phone/3G signal in most parts of the building. The main hospitalWiFi networks are secured, but there is a BTOpenZone unsecured network in the cafe.
The Starbucks on the corner is actually a lot better than the hospital canteen's. But you're better off going to The Espresso Room across the street if you have excess disposable income.