So I don't have any power going to driver side speaker once it passes through the door, and hence, will need to replace the wiring.

Problem is, I can't seem to figure how to get new cable through to the inside of the door.

Does anyone know the best way of achieving this?

I've searched other threads but wasn't too helpful.

You could take the door off which would be much easier than trying to put a wire through the harness.
You DON'T have to unbolt the door but you can lift it off without much difficulty. In between each hinge is a black plastic cylinder looking part and these can be taken out and then able to lift the door off the door hinge pins. It would be wise to have someone help you do it and also might be a good idea to have an old tyre to put the door on after you have lifted off the hinges.

Here are a few pics from another forum which should explain it better. Once you have the door off and the wiring unplugged, you should be able to get at the wires MUCH easier and thread them through your current harness. I hope that helps?

Door Hinge Pins