Welcome to the forum!
First, the image is outside the page border (far outside). That might explain some of the trouble you're having. I think the DXF conversion would convert it, even though it's outside the page border (although I don't know that for a fact), but you might not have looked outside the page border.
Also, it's huge (approx 2900 x 4600 px). So it could possibly be in the DXF file you made, but you just don't notice it, because it's larger than your screen. The image consists of a few blue lines, so it wouldn't be hard to miss them, if one of the lines doesn't pass through your screen. Maybe try scrolling around? Or zoom out?
I would start all over, and when you open the SVG in Inkscape, make sure the drawing is inside the page border, at 100% zoom. Then try the conversion again. You can set the page border to fit the drawing perfectly this way:
- Ctrl a (select all)
- Document Properties > Page tab > Resize page to contents
- File menu > Save As SVG (this way, if something goes wrong, you don't have to start all over again)
- File menu > Save As DXF