Actually, I didn’t need to scale it to fill in the faces. The main problem was that the three elements of the logo–the filament ball, the name Filament World, and the abbreviation .NL–were all in separate groups. You can’t really operate on the geometry in a group unless you either enter the group’s context (double-click to edit) or explode the group.
In the picture below, I separated and exploded the groups, made faces using the method you described, namely intersecting a rectangle with the logo linework, repaired a few minor errors such as Anssi mentioned, colored them, regrouped and re-superimposed them. I gave the text elements some thickness to prevent z-fighting with the background group. Thus:
-Gully