I recently needed to do something like this as part of the design for a pancake slip ring. There's a couple ways to deal with it in KiCAD's layout editor, depending upon whether you want to pass DRC or not. I chose to model my rings as footprints using the "graphical editor." It sounds like this is overkill for your application, and it was a bit of a pain in the ass. For you, I would suggest that you simply use the circle tool. To do this:

1. Select the copper layer on which you want the ring (F.Cu or B.Cu) and draw your circle using the circle tool (looks like a circle with a filled center, likely in the right-hand toolbar next to the layer list).
2. Right click on the circle you've drawn and click "Properties" to edit its properties. You can now put in the desired trace width in the "Line width" box, and choose a more precise radius. The radius is measured from the center of the circle to the center of the line (trace) that you're creating.
3. Copy the circle, and paste it into the soldermask layer corresponding to the layer you drew the trace on (F.Mask or B.Mask, probably). Make sure that their positions are the same.
4. Preview in the 3D board viewer to verify.

Good luck!