[Inkscape-user] Fill areas with color

I have a drawing like the attached compass.svg. I want to color it so that it looks like the attached compass.png. How do I do that in Inkscape?

Using the Paint Bucket Tool doesn't work well since it doesn't handle the sharp corners correctly, I get something like the attached SharpCorner.png (after removing the stroke of the created paths), when zooming in as much as possible.

I think that the problem is that the Paint Bucket Tool is bitmap based. I would rather need a vector based fill tool. Is there any such possibility in Inkscape?