How to find specific IP address within LAN?
So the router is the only DHCP server, right? It is the only device giving out IP addresses, and the switches (including the time-capsule) just redirect the traffic (i.e. bridge mode), right?

If I am not mistaking, the client list in your router should provide you with the list of all connected devices with mac-addresses and ip-addresses (and sometimes device names). Try and see if the mac address of your device is in there.