I'd just get a shallow, round, flush-mount light that would attach to the junction box in the ceiling. This one even has a motion sensor so the light would turn on when you enter the closet. Another example.

Some states regulate what kinds of light fixtures you can use in a closet. Check first, then decide. If you use LEDs, "bright white" is a less-red color than "soft white". That's useful if you are matching items or trying to distinguish black from deep navy blue.

I have a long narrow wall closet so we put in puck lights that are LED and on batteries....I just replace the batteries once a month....not idea but efficient, and they are motion sensor which is just perfect.....you for alot of light .

I have a 5 head LED track light all 4000K bulbs I can even aim the heads to see something clearer if I need to. A single light will not work in an 11' long space I aslo have no ide how you can have hanging space on both sides of a6' space and be able to have walk space . Maybe I just misunderstood the layout 4000 K mimics daylight and IMO is the best for getting true colors . I have a big window in my closet and have found at night the colors do not change with the 4000K

This Good Earth Lighting Plano 5 CCT 1-Light 17-in Matte White LED Flush Mount Light ($75) emits 2200 lumens, equivalent to around 3 typical recessed lights. It has adjustable color temperature and can be dimmed.

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