Welcome to the forum!
I think you might be asking for centerline trace. Although while it will give you a thin path, I don't think you'll like what happens where the lines intersect. If you want to try, you'll need to use a different trace engine, because Inkscape's Trace Bitmap doesn't offer it. You could try:
http://www.roitsystems.com/cgi-bin/autotrace/tracer.plI can't think of any easy way to use that trace result, and change it into a single, closed path. Because you may have noticed what happens when you trace an image. Select that result with the Node tool. Then you can see that it's not just a thick or wide, single path. It's a filled, compound path.
There might be a way to use Trace Bitmap to make that image look thinner. But it will never be a single closed path. It will always be a filled object.
Or you could probably use Path Inset on that trace result above. That will make it thinner too. But again, it won't be a single path. It will be the same filled object, except thinner.
Or you might be able to use that drawing as a guide, to draw your own spiro spline. You can trace over it (zoom in so it's very large, for best results). Change the color, so you can see it on top of the black one, and just tweak the nodes with the Node tool, until it matches very well.