You could do this as an outside corner, with just a little piece of rail.
I did this with some baseboard right beside a closet (that I didn't want to put trim around), and it turned out well. (Taking this picture also reminds me, I never did quite finish cleaning this all up).

The hardest part of doing this is cutting the edge piece so you get a perfect corner.
- Cut the piece (usually there's a scarp piece around that will work). Go longer rather than shorter.
- Use wood glue or a construction adhesive (like No More Nails) and a clamp to attach it, with the edges meeting. Let it sit overnight to dry.
- Sand off any excess glue that came out.
- If needed, sand down the back side so it's flat (if you cut it a bit too long). A couple seconds on a belt sander is perfect.
That's really all there is to it. In your case, you may want to also sand down the front corner a bit so it's smoother, rather than sharp and likely to poke someone.