  Let $M$ be a 3-connected matroid, and let $N$ be a 3-connected minor of $M$.
A pair $\{x_1,x_2\} \subseteq E(M)$ is $N$-detachable if one of the matroids
$M/x_1/x_2$ or $M \backslash x_1 \backslash x_2$ is both 3-connected and has an
$N$-minor. This is the second in a series of three papers where we describe the
structures that arise when it is not possible to find an $N$-detachable pair in
$M$. In the first paper in the series, we showed that, under mild assumptions,
either $M$ has an $N$-detachable pair, $M$ has one of three particular
3-separators that can appear in a matroid with no N-detachable pairs, or there
is a 3-separating set $X$ with certain strong structural properties. In this
paper, we analyse matroids with such a structured set $X$, and prove that they
have either an $N$-detachable pair, or one of five particular 3-separators that
can appear in a matroid with no $N$-detachable pairs.
