|Did that external drive ever work on that computer with that installation of Windows? |
Is the USB port you are using on the desktop computer functional? Try a different device in it to verify.
Another possible cause could be this. USB controllers support at least TWO USB ports. If the sister port has hardware with a high current draw connected to it the controller can't supply enough power to the USB drive. Solution if to rearrange the way devices are connected if that is the case.
Your 2.5" external is USB powered so that is a strong possibility.
If you have an internal card reader that complicates matters even more because they are usually connected to a controller as a THIRD port.