Replace USB port on logic board Mac mini late 2012
It's possible but difficult, especially if you aren't experienced with soldering electrical devices. 

The other problem is that your USB ports may be fine. For example, the problem could be with the USB controller on the logic board (especially if all of your USB ports stopped working!)

While your question states you have tried all non physical solutions, that doesn't actually specify what you've tried. So I'd recommend you reset the System Management Controller (SMC) and test your hardware.

Reset the SMC


Shut down your Mac mini
Unplug it from power
Wait 15 seconds
Plug the power cord back in
Wait for an additional 5 seconds
Press the power button to turn it back on


Run Apple Hardware Test

Your model Mac mini uses Apple Hardware Test. To use this, follow these steps:


Shut down your Mac mini
Restart your Mac mini
Press and hold the D key before the gray startup screen appears.
After a while, Apple Hardware Test (AHT) will start.
When prompted, select your language and click the right arrow. 
When the AHT console appears, you can choose to run Basic tests by clicking the Test button. However, I suggest you select the "Perform extended testing" checkbox before you click the Test button.
Your test results will appear in the window in the bottom-right of the console.


Note 1: that the extended test will take some time. Take a note of the results and report back. 

Note 2: If pressing and holding the D key at Step 3 doesn't work, start again at Step 1 and, at Step 3 press and hold both the OptionD keys instead. This will try and run diagnostics from the internet instead, so you will need to allow more time for it to complete.