I'm trying to break one filled path (a continent) into pieces based on the position of some other paths. I am having issues using the Path Division tool. I have combined the set of paths to divide by into one using Ctrl-K, as seen in the following image.

First, I select the path to split by, then the shape underneath (in this case the white continent). I then go to Path -> Division. However, the result is not as expected. I've shaded the regions to highlight the problem.

The lines seem to be shorter than they were drawn and are consequently not splitting properly. Does anyone know what the problem could be? All help is very appreciated.