If you're using Win9x, go to Start, Run, and type in "winipcfg". When the IP Configuration dialog pops up, select your network card in the drop down list (it may be the only item in the list). The hit the "more information" or whatever button it is at the bottom of the dialog. This will list all the available information regarding your network settings as they were set up by DHCP.
From this you can get the "gateway" address, which is the internal IP of your router. You can use that to connect to the router and make any changes to the configuration. Optionally, you can ignore the DHCP problem and simply reproduce the configuration of this machine on the second machine, changing only the IP address. It will use the same gateway, subnet mask, and DNS servers.
I'm not sure though that the tech would have configured the router to ONLY allow one machine via DHCP. In fact I'm not sure it'd be intuitive how to do that, and I think that Rogers probably would find it not worth the bother of getting the techs to do that, unless they're even more off on the idea of making people pay to use more than one machine than American cable companies. Most cable/DSL routers are essentially plug and play. They come with a default configuration which allows them to get a dynamic IP from the cable/DSL provider, and provide dynamic IP addressing for the internal network. Unless you saw the tech actually going in and configuring things in the router (via telnet or a web based config), they probably didn't do anything to make it only work with one machine. What is possible is that the "router" isn't actually doing any routing, but is just bridging the connection, so when you try to get an address on the second machine, it's actually trying to get a second address from Rogers, who won't allow that.