You have a computer running Windows XP. You want to transfer all your files to an external drive to allow an upgrade to Windows 7. This machine won't connect anymore to a network with another machine and won't go online and it will not recognize either a thumb drive nor an external drive both of which are USB items.
However, a mouse and keyboard are both USB items also and they both work. This started all at once and you have checked out both a thumb drive and an external drive which work on all those other machines. If there isn't a reasonable fix for this problem, you can swap hard drives with another reserve machine.
What is best way to handle this issue you ask? If your just trying to get your data off that computer then try booting it into Windows safe mode. You can also try booting it into "Safe mode with networking support" if you need network access.
Try removing any unnecessary USB devices when you boot it into safe mode. Try one device at a time ie. your thumb drive first and then your external drive. Hopefully, in safe mode you will be able to copy data over to your external drive.
You can also try selecting "Last known good configuration" from Windows safe mode advanced options menu. That may fix your problems. These instructions were written for a Compac Evo running Windows XP.