Hi,
The problem you are having is routing. I don't know how much you know about IP but basically there are public and private IP ranges. Public addresses are the ones give out by the ISP. When you use a router, you have both a single public address and then a private address which any other devices on your home network talks to.
NAT is the function that converts your local Private IP address into the public assigned one, effectively so you can have multiple private IPs using a single public one.
Your PC is configures with a standard and wireless lan cards effectively it has two networks installed. Your xbox is connected to one of these networks and the wireless another. when your xbox tries to connect to xbox live, it effectively is trying to connect to a public ip address but has no route out onto the internet.
when two devices are on the same ip range for example 192.168.0.1 and 192.168.0.2, there is no routing involved as there on the same IP range and physically on the same network. If 192.168.0.1 tried to connect to any address that is not on it's own network, it uses a default 'gateway' to attempt a connection.
Now, your pc's gateway will be the routers private ip address, but without configuration, the xbox's connection has no gateway, so it will be unable to connect.
the way you should configure the pc is either manually setting the IPs ranges or if your using XP, internet connection sharing will do it for you.
as an example this is how it could work. i don't know your addresses so change them to whatever you use already.
Router - 192.168.0.1
PC Wireless - 192.168.0.2
PC LAN - 192.168.1.1
PC LAN gateway - 192.168.0.1
Xbox LAN - 192.168.1.2
Xbox gateway - 192.168.1.1
try it then, good luck
Router