I think it's a good idea to add sconces on either side of the mirror for the best lighting possible. The small lamp is a nice stopgap, but as I see in the photo where it's on, it concentrates light so much on one side that the left side of your face will be thrown into shadow. Adding sconces will also leave all of your table's surface open for other things!

Wall sconces would be really nice, aesthetically and functionally.

Do you think I could also consider using a picture light that spans the top of the mirror? That way I would only have one cord hidden behind the mirror instead of two cords running down the sides of my mirror?

As a place to put on makeup, you really should have all three! :-) Top only will give you down shadows; have someone hold a lamp up there and see how it works.

How about something like this on each side of the mirror.

I'm with Maureen. I don't think I the sconces will help with makeup. I vote against the evinces. I like the elegant look you have already created