  We calculate the $E1$ breakup of the $2n$ halo nucleus $^{11}$Li in Halo
Effective Field Theory (Halo EFT) at leading order. In Halo EFT, $^{11}$Li is
treated as a three-body system of a $^{9}$Li core and two neutrons. We present
a detailed investigation of final-state interactions (FSI) in the
neutron-neutron $(nn)$ and neutron-core $(nc)$ channels. We employ Moller
operators to formulate an expansion scheme that satisfies the
non-energy-weighted cluster sum rule and successively includes higher-order
terms in the multiple-scattering series for the FSI. Computing the $E1$
strength up to third order in this scheme, we observe apparent convergence and
good agreement with experiment. The neutron-neutron FSI is by far the most
important contribution and largely determines the maximum value of the $E1$
distribution. However, inclusion of $nc$ FSI does shift the peak position to
slightly lower energies. Moreover, we investigate the sensitivity of the $E1$
response to the spin structure of the neutron-${}^9$Li interaction. We contrast
results for an interaction that is the same in the spin-1 and spin-2 channels
with one that is only operative in the spin-2 channel, and find that good
agreement with experimental data is only obtained if the interaction is present
in both spin channels. The latter case is shown to be equivalent to a
calculation in which the spin of $^9$Li is neglected.
