For some reason, the IP addresses of devices connected to my second router are invisible to the system connected to the first router. For the second router however, I can view the systems connected to the first. Also, I can't view the router's index page from a computer connected to the first but can view both from the second. I suppose there's probably a networking fundamental I've missed here.
It sounds like you plugged a cable from the LAN side of the first router to the WAN side of the second router.
An easy way to test. Are the first three digit groups of the IP address the same on all your computers (regardless of which router they are plugged into)? If they are different then I am 100% sure that what I said is the case. Plug the cable connecting the two routers to the LAN side of each.
Before you do that though, you will need to go into the index page of the second router and turn off DHCP and statically set the second routers LAN IP address to an address in the range of the first router. Effectively, you are disabling the router part of the router and making it a network switch which is probably what you need anyways.