It turns out that the problem originated with the statement by the previous installer that "Just plug your router into the back of the new modem and everything will work." As it turns out, this was *almost* correct.
The new cable modem is *also* a router. So when you plug a router into the back of another router, you get *two* home networks that don't talk to each other. The eventual solution was to reconfigure the cable modem's router so that it was on the old set of IP addresses that I use around the house. Then I could reconfigure the existing router and WAP so that the router no longer tried to be a DHCP server (making the cable modem happier) and that both of them were on static IPs outside of the DHCP range (because trying to find them to log into them with dynamic IPs is like shooting fish in Lake Superior).
I have ordered some new network cables -- because, as Gretchen observes, *everything* that I do requires another cable -- but it looks like I now have a very fast little connection, all on the same network.