Welcome to the forum, Bob.
The Dremel & diamond wheel will work. But because of the angle the wheel will be tilted at, you likely find it easier to work from the top and bottom of the tile to cut it out. It’ll leave a ‘V’ shaped channel. The tile file will likely do more to scratch up the neighboring tile than to smooth the offending piece. If you did this, you could grout over the shallow channel when done.
Or you could leave the broken tile as-is, and use a triangular piece of 2” KerdiBoard with a channel routed out to straddle the old tile. Use KerdiFix to adhere it to the surface of the very clean tile. Then tile over the top, front, & bottom of it.
Or you could get a triangle of Corian or something similar (check your local countertop shop for scraps) and fabricate a corner shelf with a couple of cleats under it. You’d have to remove the broken tile, but you’d have wiggle room to not be so super neat in removing the broken piece.
By the way, I can see that the corner shelf was natural stone. And such a skinny piece of natural stone was bound to break at some point. Most tilers will at least double up on the tile thickness if they’re going to use it for a corner shelf. And if it’s meant for a shaving toe ledge, even that is too wimpy for many stone types. Don’t feel bad about breaking that.