If both network's LANs are using different network numbers, this should work. If both networks happen to be numbered the same, one will have to be re-numbered on the LAN side.
"Your Router" --- Network A's router that connects to A's ISP, with a LAN gateway of 192.168.1.1 and a netmask of 255.255.255.0. (whatever it really is)
"The Other Router" --- Network B's router that connects to B's ISP, with a LAN gateway of 192.168.5.1 and a netmask of 255.255.255.0. (whatever it really is)
"The Spare Router" --- That extra router.
On the spare router, disable DHCP client on WAN and and DHCP server on LAN; both interfaces will have to be configured manually. Also disable NAT on the spare router.
Find an unused IP address on your router's LAN network that is not in the DHCP pool. Often, there are a few extra addresses at the bottom of the network that aren't handed out by the DHCP server: If your LAN gateway is 192.168.1.1, usually 192.168.1.2 is not automatically handed out. In any case, find an unused IP address on your router's LAN, and manually configure the spare router's LAN to that IP; I shall assume it is the .2 number. Also set the spare router's LAN netmask and other network attributes to match your LAN's parameters.
Do much the same on the other router's LAN: Find an unused IP address outside of the DHCP pool. If there are no addresses outside of the pool, you'll have to reserve one so it's not assigned to another host accidentally. I shall assume that IP address is also the .2. On the spare router, manually configure the spare router's WAN port to 192.168.5.2, and its netmask to that of the other router's LAN.
Setting the spare router's DNS and gateway is not absolutely required, but setting it to the other router's gateway would be OK.
The spare router should then have a WAN interface numbered 192.168.5.2 and be connected to the other router's LAN. It should also have a LAN interface of 192.168.1.2 and be connected to your router's LAN. The spare router should be able to ping hosts on your LAN, as well as on the other LAN at this point without any further configuration.
On your router, create a static route:
192.168.5.0/255.255.255.0 gateway 192.168.1.2
On the other router, create a static route:
192.168.1.0/255.255.255.0 gateway 192.168.5.2
Hosts on your network should now be able to reach hosts on the other network. Remember that NAT must be disabled on the spare router.