If the drive is a 2.5" Passport, then it will have the USB port on the hard drive's PCB. Your only real option then is to have someone resolder the microUSB connector for you, even if they have to attach it with wires (eg wirewrap wire).
OTOH, if the external drive is a My Book, then it will have a separate USB-SATA bridge board. This then gives you two options:
(1) If the drive is an Elements, then the data will not be encrypted, in which case you could connect the HDD directly to a SATA port on your computer's motherboard, or install it in a third party enclosure.
(2) If the drive is an Essentials, then it will incorporate 128-bit AES hardware encryption, even if you have not set a password. This will mean that your data will appear as gibberish if you use one of the options in (1). In this case your only solution, other than resoldering the connector, would be to purchase an identical product of the same capacity and swap the bridge boards.