Ok. Here goes some images more then.
First thing is always Apply Scale, Rotation to ALL. Even if you know that’s not needed - do this. Wont harm anything. Same goes to Recalculate Normals, but that’s another story.
So we start with Curve to follow. Mesh Circle is a good candidate, so tear it apart move half of vertices away from center, Mirror or just F connect halves back. Convert that circle to Curve. Alt-C, works forth and back. Done.
Coil - add Circle again ( all “Add this and that” happens on Grid centre so far - Shift-S - Cursor to Centre); less vertices this time - T-panel, some 6 to 8 is fine. Scale it down (see above, it’s time for the First Thing) and in edit mode move vertices away from Centre.
Add Screw modifier so that there’s just 1 turn.
If Scale and Rotation is applied to both objects it’s time to fiddle to get them work together. One needs to be rotated against other to get it in proper position. Again, Ctrl- A, Rotation… The rest is in the pictures.
Gap happens because each imprecision for 1 turn summs up in the end. Needs to be little bit adjusted unfortunately.
Nurbs Paths and Objects on them work just fine - see First Thing.
File attachments could be saved on pasteall.org and link posted there.