Instead of using LPE's you can use the Bezier/Line tool and select a tapered shape as the shape for the line.
The process is described here;
A little "Spiro - Tutorial";
(You can save this SVG file and open in your own Inkscape -
right-click the image (Above) and use "save link as" (Firefox) or
"Save target as" (Internet Explorer).
Good Luck.
RGDS
Ragnar