It is possible to assume that $\ell$ is horizontal and that it does not go through any intersection of lines in $\mathcal{L}$. Also $\ell$ will intersect $|\mathcal{L}|+1$ faces. We will look at what is above $\ell$. All half-faces will consist of one chain of edges going up from the left and one from the right (or just one chain for the two faces which $\ell$ intersects in a half-line). These chains will either not meet, or will meet in a top-most point, or there will be a line parallel to $\ell$ at the top. We will call left edges those that are in the left chain, but not those that are either infinite or meet the right chain at a point - these, and a possible edge parallel to $\ell$, we will call the topmost edges. Similarly for right edges. Now, we observe that any line can be a left edge at most once (otherwise if it gives left edges $e$ and $e^{\prime}$, with $e$ lower than $e^{\prime}, e^{\prime}$ will be separated from $\ell$ by the topmost left edge of the face of $e$ ). Thus, there are at most $2|\mathcal{L}|$ left and right edges, and another possible $2|\mathcal{L}|$ topmost edges. Same for the edges below $\ell$. Yet $2|\mathcal{L}|$ edges are counted twice, once for above and once for below. It follows that the total number of edges is at most $6|\mathcal{L}|$.