First, you do not need a separate modem - you can connect your laptop wirelessly to the router and share the internet connection with your desktop.
Second, I'm not sure what you mean when you say "I test the connection through the Router config window it just fails". What fails? Can you connect to the router? Does the router connect to the internet?
In any event, you will be better off by setting the desktop to obtain an IP address automatically rather than giving it a fixed address.