You need to figure out how water is getting into the studs. Once that problem is solved plain wood wil be fine. You should sister on new wood about 12 inches ablve the rot, fasten with screws (nailing can cause plaster damage), and then remove as much of the rotted wood as you can. If the wall is weight bearing you should sister the other side after removing the rot. Matal brackets (Simpson Strong tie) to attach the ssters to the bottom plate work well. You need to chek the bottom plate for damage also. They can be more work to replace depending on how extensive the damage is, but still not really hard. I am just finishing up a basment repair. The bulder back around 1972 used non-PT wood for the bottom plate of a simple curtain wall in a basement. The plate has serious rot prolems. After checking the attachment to the joist above, I sawed through the nail attaching every stud to the plate, then hammered tha plate out (it was rotten enough to break up pretty easily). A new PT plate (below grade maonry contact), brackets, some stainless fastners for the bracket to the PT and regular galvanized TECO nails for the bracket to stud and everythig is back in shape.

Thanks! I thought about trying to shave the cabinet, but it only has thin plywood sides to start with so was worried about doing damage there. The block is basically just a piece of wood (probably the piece of stud that was cut out) nailed to a horizontal 2x4 that is in turn nailed to the remaining studs. It formed the left edge of the original cabinet. We could also take it out and just move it, but the lathe is nailed to it so that seemed to have the potential to do a lot more plaster damage. (We have a clear inch on both the block and the stud that could be shaved with no impact to lathe, probably because of the way the original cabinet was set in place.) The plywood stabilizer is a great idea, though--thanks!

It's a basic two gang box. The sides are flat, so this should work - right? The wings that tighten with screws probably won't work because the wall is too thick - they can't slip behind it. I have thought about putting a magnet on a string and going fishing for the metal thing that held the box in previously.

It sounds like you have what you need, and that you just need some practice. Any corded drill (or cordless) can handle this job. It sounds like you have standard, inexpensive bits which are usually high speed steel (HSS), and those are fine, too. You may want to get some better quality bits (like titanium coated) in the future. Unless your walls are actually concrete, you have no need of specialty tools, like a hammer drill or masonry bits. Impact drivers are wonderful specialty tools for driving fasteners into tough materials, but it would be the wrong tool to use on this project. (It would be nice to have a cordless drill-driver if you are driving screws, but you don't absolutely need that, and you need to master the use of a regular drill first.) Plaster drilling dulls bits faster than using them in wood, so you might want to have some extra bits on hand. And even though I have been using tools for over 50 years, I, too, occasionally break small diameter bits, so I keep cheap spares around. While I tend to buy high quality power tools, when it comes to disposables, like bits, I look for bargains since I do a lot of drilling; for your drilling needs, you might want to look at a place like Harbor Freight tools: Drill bits You can get decent bits there for cheap, including things like a set of half a dozen 1/16" or 3/32" or 1/8" bits for a couple of bucks. They have stores in many areas of the country and you can buy online. Too many homeowners have one set of bits that get dull early on and never get replaced, so I suggest you pick up some extras and toss them out when they quit performing well. Good luck, and tell me what happens when you try this again. Do you think it would help your skills development if you practice drilling into a scrap of wood first, just to get the hang of handling your new tool? I also assume that you have a standard variable speed drill, and it would be better to use it at a slower speed when going in to plaster. And did you confirm that the forward-reverse switch was in the correct position? :-)