My old Netgear router would show any devices attached, whether using DHCP or not. My new Cisco one only shows DHCP clients connected.. Anyway, static IP or dynamic IP shouldn't matter. If the router sees it, the connection seems to be physically ok. What about your subnet mask? Normally you should have 255.255.255.0 on *all* devices on your network - router, MiamiDX, PC. Then the IP addresses for everything should have the first three parts the same - e.g. 192.168.1.x, and only the x should be different for each device.
Try set the Amiga to have a static IP first off as there are some minor issues with DHCP support under certain circumstances using MiamiDX. Give it an address outside what the router assigns (typically this might be 192.168.1.1-192.168.1.51, so try 192.168.1.100), set the subnet mask correctly and you should hopefully be able to ping the router...
