Hi @tnskycop ,
Here is a link to a [https://www.youtube.com/watch?v=STo-WlCHYh4|video] which shows how to disassemble the camera so that the USB port can be removed.
As you say that the USB port is physically broken, here is a link to a [https://www.ebay.com/itm/3pcs-Mini-USB-Charging-Port-Jack-Connector-GoPro-HERO-3-3-Silver-Black-HERO-4/131905270642?hash=item1eb62ab372:g:Zv4AAOSwMtxXq91g|supplier] of the replacement part. It is not a recommendation to use them. It is shown to give an idea of the cost of the part. There are other suppliers online that may suit you better. Just search for ''GoPro Hero3 Black USB port part'' to get results.
To remove / replace the port you'll need SMD (surface mount device) soldering tools and expertise.
If it all seems to daunting to do, especially the SMD soldering, you could try dis-assembling the camera and once you have the replacement part, contact a reputable mobile phone or electronics repair service who should be able to professionally solder the new port onto the board for you.