You'll need a power point near the phone socket.
If that's OK, buy a Netgear DG834G wireless router from Savastore (they have them in stock), and set it up using your new laptop and the cable that comes with the router. Once you've done the initial setup, which is simple, and fully explained in the router's instruction sheet, you can disconnect the laptop from the router, leave the router running downstairs, and use your laptop all over the house.
If you get a PCMCIA wireless adapter for your old laptop (or an external USB adapter) your brother can share the internet connection with you, and if your mother orders her new PC with an internal wireless PCI card adapter she can surft the net too.
Great, isn't it?