Boolean operations only work on closed regions/geometries — the magenta lines indicate unclosed/unjoined lines:
You need to convert them into a closed region:
In order to do this it will be necessary to draw in the gaps so as to close them:
Note that at this point, it is just easiest to use the polyline tool to just draw the entire thing:
Doing this however, shows that things do not line up as well as one might wish:
so arguably one would want to adjust things for a better result:
Note that we already had this discussion: