Seems like you joined the wrong end nodes, one from the spiral center and one from the end. I didn't get the line while following the steps although workflow in Inkscape is a bit different. Here's how I did it:
draw a spiral, adjust its shape to your liking, remove fill, add stroke
duplicate the spiral Ctrl + d
hold Ctrl and Shift while dragging one of the corner handles to resize the second spiral (Ctrl will keep its ratio and Shift will keep it centered)
make sure you have these two snapping preferences set under Document properties: What snaps - nodes, Snap to objects - snap to nodes (in the development builds there is a snapping toolbar that makes this even more easier and there are onscreen tips)
zoom in on the spirals center, grab one of the spirals and move it so their centers (end nodes) snap together
select both spirals and combine them into one object Ctrl + k (in Illustrator this step isn't needed)
now you can join end nodes using Node tool, select the two center end nodes that are snapped together and use Join selected end nodes button from the toolbar
select the outer end nodes and use Join selected endnodes with a new segment
make that segment into a straight line if necessary (also a button on the toolbar)
now you have a closed shape and you can set the fill