Electrical – Old Work Box for Vanity Light

I see a couple of issues. The first is that your three bathrooms are not wired properly, and you should take this opportunity to fix them! You should not have outlets in three different bathrooms all on one circuit. Each bathroom should have its own 20 amp circuit. The reason for this is that hair dryers, curling irons, etc. use a lot of juice.

Your second issue is going to be the ability to "T" off of an existing cable at some point below the existing outlet. Assuming that the cable actually does run down the wall below that outlet, I would imagine it runs in a fairly straight line without a bunch of slack, right? Unfortunately, in order to "T" off of it, you would have to cut that cable and then have enough slack to pull 6 inches or so from each side of the cut into a new box, where you would join those two ends with your new cable. It's highly unlikely that your wire has a foot of slack in it that you could use for this purpose. If it did have a foot of slack, then you would be better off just pulling it through the actual hole for the new outlet and connecting the two ends (from the top and bottom of the cut) to the screws on your new outlet - no need for a new cable or a secondary hole anywhere.

All that said, here's how I would go about accomplishing what you want to do without cutting any unnecessary drywall:

• Turn off the power and remove that outlet in the picture from the box.
• Put a crowbar or something against the inside of the box, and hit it with a hammer a couple of times - just enough to make the box start to lean away from the stud it's attached to so you can see the sides of the nails.
• Slide in the blade of a reciprocating saw and cut off the nails.
• Trace the old cable from the outlet back to the outlet that was feeding it power.
• Disconnect the cable from there and pull it out of the wall if possible. If it's fastened inside the wall, just cut it off and leave it in place.
• Run a new 12-2 cable from your breaker box to the new hole you cut in your drywall for the outlet under your vanity. Run another 12-2 cable from that location to the hole your old outlet box came out of.
• Get a pair of "old work" single gang boxes. These are designed to slide into the small rectangular hole in your drywall and clamp onto it from the back with no need to cut giant holes that will require taping and repair.
• Pull the cables into those boxes and fasten the boxes to the wall.
• Install a new GFCI outlet under the sink using the directions in the package.
• Install a new 15 amp outlet where the old one was located above the sink.

One possible investigatory tool is a "toner" set - Fox & Hound or some other brand - While they are are more often used on telecommunications cables, some do claim to work on power wires. User reports vary, possibly due to different toner sets working more successfully than others with power wires. They work by one part applying a small electrical signal to the wires, which the other part of the set can pick up without having to actually touch the wires. If you can get a tone on a circuit (Disconnect it at the breaker before hooking up the toner - assuming you understand the risks of and precautions to take when working in the breaker panel [scratch that - it's the landlord's problem - leave it alone]) and then you find a place in the wall where tone stops, rather than going off in any of the possible directions at that point in the wall, it might mean you've found the break. Check both sides of the wall, etc. to increase the odds of success. Try connecting the toner at one of the dead outlets and tracing backwards, as well. If they both stop in the same place coming from different directions, that's a good place to knock a hole in the wall.

Before any more fail, map whatever circuits still work, and perhaps turn off any you can't find out what's connected to (paying attention to things that might not be instantly obvious like furnace or boiler, well pump if any, sump pumps, etc...)