How do I finish the open end of a chair rail?
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.