Need help solving this tricky connect paths problem!

Not a script, but the 'Concatenate' plug-in would be useful for doing this manually.

How often do you need to do this? I think that a script which would work properly may be more work than is worth the time.

I'm sure something like this could be done, but the real problem would be what would the next group of lines that you run the script on look like? Would a script that works for this circumstance work properly for another shape?

Here's some parameters to think about:

How do you determine which line segments (or endpoints) in the document need to be connected? All points in the document, or will you be working with a selection?

Once you determine the points (eliminating all that are already connected), how do you want them connected, and which point do you start with? Connect to the closest available point would seem what you want, but depending on how you determine which point to start with could give you different results.

You might want to loop through all eligible points, and save the distance to every other point into an array. Then, loop through connecting the 2 closest available points until you've connected every point.