So: telephone appears to work again.
It appears that the problem was that the handset didn't know what number to broadcast or something. When I set it to use our ISDN connection's main number, that solved that.
However, I wanted it to advertise a secondary number... stupid habit from back when we had two separate telephones and I wanted both of them to ring when the main number is dialled but to advertise separate numbers on outgoing calls so that I could tell which one was used on the itemised phone bill.
I got Stella's computer connected with a LAN cable, downloaded Ad-Aware and Spyboat Search & Destroy and Avira (all of which found several different bits of malware, including one that had infested the system tray, displaying as a little shield with a cross and popping up message balloons occasionally telling me that spyware had been detected and I should download software from some website or other). However, I wasn't able to manage to connect to the Internet via WLAN with her laptop.
Today, I tried my company laptop, and that worked just fine. I fiddled with the router's settings through its web interface (disconnecting myself in the process when I changed the WLAN SSID and set broadcast to off... fortunately, I managed to configure the computer's WLAN settings to get connected again).
I also found a place where I could enter my telephone connection's MSNs... and once I had done that, the phone seemed to work even when I set it to use the "handheld" MSN rather than the main one! So yay!
And it still rings when the main number is called. So all is well.
My desktop computer is still busted. My brother-in-law, whom Stella spoke with, said that it sounds as if the power supply is fried. She'll take it into him to look at next week; we shall see what will transpire thereafter. (If it's only that, that should be easy enough to remedy.)