I only have one line selected, I click the other one and it just selects that one. I made them using the Bezel tool, does that matter? I tried using both the regular pointer and the node pointer.
Okay turns out I was doing that right, but something is wrong with my "division" action. I select both lines and choose "division" but it just makes the interior line disappear.
Maybe your line isn't long enough? As in, are you sure it does touch and exceed your figure's lines in two different places (like in #3 point in sixth post in the thread)? :p
Try redrawing the line completely and/or moving it to a different location, then try again. It doesn't happen very often, but occasionally if the positioning of the line and the territory line up in just the right way, it won't work no matter what you do. I can't explain how or why; just give it a shot and see if it works.
If even that doesn't work, you might want to give your SVG to an experienced mapmaker and see if they can fix it for you.