Selecting ALT-J will try to auto-join these sections into a single, closed path shape. You can see this change indicated by a direction change in the “marching ants” marquee of the selection. You can find open shapes by selecting ‘Edit’→’Select open shapes’ from the menu.
Why won’t it work here? You will need to edit the line if what you want is for the segment to follow in the same arching path, but it does work.
I don’t find the exact same section in the file I have but it does have circles with gaps, so I will use that in my example.
I draw a line segment snaping to the ends of the gap, then enter the ‘Node Edit’ mode.
Grab the center of this new line segment and drag down to match the arching shape. I drew another circle and used it as a template to help with visual alignment.
Again, selecting ALT-J will try to auto-join these sections into a single, closed path shape.