I had planned to stick the spindle directly through the holes in the wooden spindle-stands, but this was not feasible due to the slight crookedness of the current spindle. So I have pulled leather thongs through the holes that now serve as spindle bearings, holding the spindle right in front of the stands on the wheel side. Two wedges on the other side of the wheel stands serve for fine-tuning the arrangement. You can more or less see the whole thing on the second photo, if you click on it and enlarge it.
Panth, yes - I looked at another manuscript for the overall height, and that one set the axis consistently at somewhere between breast and shoulder height, with the accompanying long legs. And then I reasoned that it's easier to shorten long legs than elongate short ones...
Cathelina, yes, I spin with hand-spindles as well (you should be able to find some posts about that under the "spinning" tag). The wool has to be prepared differently, for long-draw, for this wheel, and thus the technique is also different. And the wheel is much, much faster than the spindle. I am planning on doing some tests soon.
It's a leather band (not a round one, though, but a square one), and I tied it together with, indeed, a tensioning string made from linen. I just knotted the string around the leather band tightly enough that it doesn't slip on its own, but it is possible to slide the knot along the leather to adjust the tension. It's neither very beautiful nor very sophisticated, but it does work!