Join only works on end points or stray points, you cannot join using a point that is in the middle of a path. Stray points look like tiny Xs, which is also what Illustrator displays to show you the centre of an object. It sounds like you are confusing the two symbols. Go to Window > Attributes and toggle the centre point option (that’s at the left of the panel). If the points disappear, then these are centre markers and are not actually points you can manipulate. Can you post a screen shot showing the artwork in question and highlighting the points you wish to join?
Thanks for the reply.
Attached is a screenshot highlighting the points I wish to join. Meanwhile, I'll try and toggle the centre point option you mentioned.
The first image does not show.
It seems that you are actually trying to join end Anchor Points, and that is done with the Direct Selection Tool as you have tried.
I believe something else is is cluttering up things. If you zoom in a use the Direct Selection Tool, you may find that there is something else there, possible duplicate paths. You can look in the View>Document Info palette, change from Document to Objects, and you can use the Layers palette to lock the swirlies so you can see whatever else might be there.
If there is more than the two swirlies, delete the rest and retry.
It seems that you wish to join the swirlies at both ends. Are attempts unsuccessful at both ends? Otherwise, you may join at one end and just select the now single open path and join again; after that, you may lock it and delete whatever is there.