How can I run wiring around an inside corner?
If you are having issues with drilling the corner, realize that you can make the job significantly more possible with an ordinary drill if you don't try to hold it level - shift the wire up or down a few inches as well as taking it around the corner, and the hole becomes much easier to drill.

As for your question about the approach using conduit, that's going to depend on the LAHJ (local authority having jurisdiction) a.k.a. "the building inspector", if you are subject to inspection. Generally better to ask them first, than to try and convince them that the internet said X would be OK...what is acceptable in one local area may not be in another, and in some cases what's acceptable in the same area will vary with inspector (probably shouldn't, but often does, even so.)