  In a recent publication, the flexion aperture mass statistic was found to
provide a robust and effective method by which substructure in galaxy clusters
might be mapped. Moreover, we suggested that the masses and mass profile of
structures might be constrained using this method. In this paper, we apply the
flexion aperture mass technique to HST ACS images of Abell 1689. We demonstrate
that the flexion aperture mass statistic is sensitive to small-scale structures
in the central region of the cluster. While the central potential is not
constrained by our method, due largely to missing data in the central
0.5$^\prime$ of the cluster, we are able to place constraints on the masses and
mass profiles of prominent substructures. We identify 4 separate mass peaks,
and use the peak aperture mass signal and zero signal radius in each case to
constrain the masses and mass profiles of these substructures. The three most
massive peaks exhibit complex small-scale structure, and the masses indicated
by the flexion aperture mass statistic suggest that these three peaks represent
the dominant substructure component of the cluster ($\sim 7\times
10^{14}h^{-1}M_\odot$). Their complex structure indicates that the cluster --
far from being relaxed -- may have recently undergone a merger. The smaller,
subsidiary peak is located coincident with a group of galaxies within the
cluster, with mass $\sim 1\times10^{14}h^{-1}M_\odot$. These results are in
excellent agreement with previous substructure studies of this cluster.
