There is no wireless on this system. The details of the LAN card as you've written them do not specify any wireless device.
Most desktops do not come with wireless devices.
Your two options are:
1) purchase a wireless device and install it.
2) connect to your router using a network cord.
Regarding your desire not to wire the connection: Desktop wireless cards generally do not have anywhere near the same range as their laptop brethren. This is usually because the antenna for the wireless radio in a laptop is able to run up inside of the screen, making it rather large and generally at least a foot or two away from anything that would block it.
In order to use your spare wireless router, the device will need to support something called Bridging, which many routers either do not support, or doesn't work even if they do support.
If you absolutely have to use wireless, make sure to get a card (or USB device) that has a seperate antenna that you can place up and away from the actual computer unit so as to extend the range, remove it away from interfering and blocking things, and just generally make sure you have the best possible option.
However, you're running DSL, which means you should be able to plug your modem in at any telephone jack in your house.
Are you certain there are no nearer telephone jacks to your computer than the one you have your modem connected to now?