For what it's worth.

  • Select all your Lines that need a Fill.
  • Clone them.
  • Join Shapes.
  • Add a 1px Outer Contour with a Mitre join.
  • Click Inset Path (Shape of Lines).
  • Cover this 'Shape of Lines' with a Box of the required Colour.
  • Select this and the 'Shape of Lines'.
  • Combine Shapes > Intersect All Shapes.
  • Discard the outer offcut.
  • Fit inner shape to original Lines.

Acorn