What's the best way to fix a mini-usb port on an external hard drive enclosure?

I do have a spare WD hardrive enclosure that may be the same physical size, but I can’t figure out how to open it as it has no visible screws.

If the connector is beyond repair then you have nothing to lose by assuming it isn’t. So get a needle nose pliers & see if you can bend the entire connector into realignment with the housing and/or bend the outer metal shell enough to allow re-plugging in the cable. It might actually work long enough to back up data.

If you get this far & it still doesn’t work, the next step involves a screwdriver and soldering iron. If you’re lucky there’s a broken wire/conductor where the connector is soldered to the circuit board, which can be bridged with a bits of wire & solder. If you’re not lucky, the connector will have to be replaced & you’ll never get the right part before the local shop opens anyway.

If it’s a true MacGyver-style emergency (foiling a plot to blow up the world, say) you could solder all 4 individual wires from the cable directly to the board—2 power, 2 signal. Good luck with that lol.