If you don't want to start from scratch by erasing the program and using BOOTP/DHCP, you're going to "sniff out" the IP address.
If you know the subnet that the controller is probably connected to (like 192.168.1.x), use a network scanner like Angry IP Scanner or NMAP to find it.
If you don't know the IP address at all, I think the best thing to do would be to connect to a local Ethernet switch with your computer running Wireshark, then watch for ARP packets when you physically connect the SLC-5/05 Ethernet cable to the same switch.
Information on Wireshark is widely available on the Internet and at Wireshark.org.
If the PLC does have an IP address (i.e. it's not using BOOTP/DHCP mode), you may also be able to use the EtherNet/IP driver to determine the IP address, since it uses a Broadcast packet to find controllers. This also relies upon the SLC-5/05 supporting EtherNet/IP, which has only been for about the past 10 years.