I'm not sure what the problem was to be perfectly honest, but I managed to work around whatever it was.
Select the path you wish to divide and use: Path > UnionKeyboard shortcut: CTRL
If you try dividing it again by following the wiki guide, just as you were doing before, it should work. Or at least, it did for me when using the file you shared.
Chuck, I could never draw each territory individually; it's just so much slower. If that's your preferred method though, that's your choice, but if you don't know about the Division method it's worth exploring.
Read up on it here: https://goo.gl/wLwDmh