KoRn indicates the hospital IT dept would have a spare and they might be happy to lend it. You could call the nurses station and they could answer your question above and you could ask them to see what they could do about getting a spare. Since the floor nurses station is well aware of the situation, I think they would be happy to talk to the right people. Your nephew's doc could also intervene.
That would solve everything quickly. When a doctor wants something done, it gets done. Asking about whether the phones are analogue or not could be an issue. I doubt 90% of the people working there could tell you what kind of phones they use. Most people just simply aren't aware of such things. If this were my hospital I would give them a USB wireless network adapter to use and get online. I have an entirely separate physical network just for public wireless Internet access, they may have the same there. It all depends on the hospital really.