Take one of the Laptops that does well as Wireless and plug the Desktop cable to its wire connection. Make sure that it is configured to use the the wire correctly and see if it works.
If it does work it means that the Router and the cable are OK and you need to fix the computer. If it does not work with a cable connected to one of the Laptops, then something is wrong with the Router LAN wire part.
-----------
As for the computer (this depicts Win 7, Vista is basically the same)
Check the Device Manager for valid Wire card entry.
http://www.ezlan.net/Win7/net_dm.jpg
If there is No valid entry, delete any Bogus Entry, and re-install the Wire card's Drivers.
Try to install the latest drivers of the card from the support page of the manufacturer.
Download the drivers and unpack them to a folder of your choice.
In the Device manger, right click on the adapter.
Click on Update Drivers, and choose the Browse my computer for drivers,
Choose let me pick from a list, Click on Have a Disk and point to the folder with the Drivers.
-------------------------------------
Check the Network connections to make sure that you have a Local Area Connection Icon/Entry, and that the Properties of the Icon (right click on the Icon) are correctly configured with TCP/IPv4 protocol in the Network Connections Properties.
http://www.ezlan.net/Win7/net_connection_tcp.jpg
Right click on the Wire card Network Connection, choose Status, Details, and see if it got an IP and the rest of the settings.
On IP status.
TCP/IP and its variables typically look like this.
Right click on the Wire card Network Connection, choose Status, Details, and see if it got an IP and the rest of the settings.
Typical TCP/IP status looks like this.
http://www.ezlan.net/Win7/status-nic.jpg
Description is the card manufacturing data.
Physical address is the card's MAC number.
The xx should be a number from 0 to 255 (all xx same number).
yy should be from 0 to 255
zz should be from 0 to 255 (all zz the same number.
The lease date should be valid to the current time.
*Note 1. IP that starts with 169.xxx.xxx.xxx is not valid functional IP.