Grounding an old 2-Wire system in my bathroom
With wiring that old the best and EASIEST thing to do is abandon all existing electrical in the bathroom (or complete area of renovation) and replace it, from point A to point B, as if it were new construction. It would be crazy to have the walls opened up and not do so.


It would be best for point A to be a new circuit run from the panel.
2nd best is tapping the exiting circuit before it enters the renovation area in an attic, basement, crawlspace or a light or receptacle in another room making that your point A.
If you must, tap the existing circuit where it 1st enters the renovation area making that your point A.


The 1st device in your new wire run should be a GFCI receptacle that's installed properly to protect the remainder of the circuit. BTW a GFCI doesn't require a ground wire to make the circuit safer. (If you're getting shocked from hot to neutral a GFCI won't help you, but if you're getting shocked from hot to something grounded it will, regardless of whether there's a ground wire or not.)

Don't ever splice outside of a box even if that box will not otherwise be used and have to have a blank cover, and whether you use the old circuit or not, don't forget to maintain that circuit if it happens to be continuing on (passing through) to supply another part of the house.