Hello Carrie. Your dressing room does need additional lighting. I can imagine how dim it is when you need to dress up in the evening.
A recessed light in the middle of the room across the two mirrors would be okay. But if your ceiling is quite low, I would recommend a diffused type (perhaps an LED panel) which casts a wider beam and creates less shadows when you look at your self in the mirror.
I suggest you replace the lamp beside the circular mirror with a wall-mounted type. The one you have is too low, it's not too helpful when you are checking your face on the mirror or applying make-up.
Placing a track light along the length of the area where you hang your clothes will be good. You can adjust the direction of the lights as well the position of the fixtures along the track.
In our store, we carry a German brand called Paulmann. They have specialty lights which you can install in your cabinets. I'm not sure if you have this there, but you might want to check. (http://www.paulmann.com/en/all/pages/light-inside-closets.html)
Lastly, just make sure to check the CRI (Color-Rendering Index) when you buy your lamps. This is the measure of the light's capacity to accurately reveal true colors of objects. It should be at least 80. It will aid you when you select your clothes.
I hope this helped.

In this traditional space with a relatively low ceiling, I think you can do some simple but pretty flush mount fixtures or semi flush mount fixtures; look at Hudson Valley Lighting. Recessed light fixtures are fine but don't offer any character.

Thanks, Joni. If you have suggestions for optimal placement, that would be great!

there are new recessed LED lights they provide a amazing light

I agree that simple flushmount fixtures would probably do the job here -- a smaller one in the closet region (1-2 bulbs) and a larger one in the larger space (2-3 bulbs). But you could do the LED cans too. Here are the two layouts I'd use.

Thanks so much, acm!

Here are a few flushmounts to give a sense of directions you might go....

Smith 3-Light Bushed Nickel Flush-mount Ceiling Fixture · More Info

WAC Lighting Dice Square LED Flush Mount, Brushed Nickel, 6", Soft White 3000k · More Info

Nuvo Lighting 60/4627 Harlow Two Light Flush Dome Ceiling Fixture With Slate Gra · More Info

Trans Globe Caged Glass 20 Inch Flush Mount · More Info

Belle 4-Light White Thread and Chrome Flush Mount With Hanging Crystals Glam · More Info

Andante Industrial Semi Flushmount Ceiling Lamp, Antique Brass · More Info

That sounds great, acm. And I really appreciate you giving me your placement suggestions on the layout. I think you're right, that 2 flush mounts will do the job. One back in the dark "closet area", and another where my mirrors are.

Thanks, everyone. I know I really need better lighting. I've been living in this house for 17 years(!) and am just now working on looooong overdue improvements. I think I'll go with two flush mount lights - I don't apply makeup in this room (I hardly ever apply makeup at all, for that matter) or do my hair, etc., so don't need that kind of light.

I want the dark back part of the room to not feel like a dungeon when I'm picking out clothes, and can use better light for picking out outfits for the rare occasions that I have to look respectable.