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From the look of your screenshot, they should join without any problem. The Join Selected Nodes button on the control bar looks like this
Or there's a key shortcut for it. They appear to be selected and ready to join, in the screenshot.
If that doesn't work, my first guess for what could be preventing it, is that there is more than one node on top of each other, such that you're trying to join a non-endnode, for at least one of the paths.
To do more than guess, we'd have to be able to see the SVG file.
Unfortunately (and for me it's always surprising) we often hear from people who have taken a job to do something that they either don't already know how to do, or don't know much about.
Inkscape is not just an art program. It's very sophisticated and complex vector graphics editor. It does so much more than art!
Oops, flamingolady sneaked in while I was typing!