The flux of neutrinos from annihilation of gravitationally captured dark matter in the Sun
has significant constraints from direct-detection experiments. However, these constraints are
relaxed for inelastic dark matter as inelastic dark matter interactions generate less energetic
nuclear recoils compared to elastic dark matter interactions. In this paper, we explore the
possibility for large volume underground neutrino experiments to detect the neutrino flux from
captured inelastic dark matter in the Sun. The neutrino spectrum has two components: a
mono-energetic “spike” from pion and kaon decays at rest and a broad-spectrum “shoulder” from
prompt primary meson decays. We focus on detecting the shoulder neutrinos from annihilation of
hadrophilic inelastic dark matter with masses in the range 4–100 GeV and the mass splittings in
up to 300 keV. We determine the event selection criterion for DUNE to identify GeV-scale muon
neutrinos and anti-neutrinos originating from hadrophilic dark matter annihilation in the Sun, and
forecast the sensitivity from contained events. We also map the current bounds from
Super-Kamiokande and IceCube on elastic dark matter, as well as the projected limits from
Hyper-Kamiokande, to the parameter space of inelastic dark matter. We find that there is a region
of parameter space that these neutrino experiments are more sensitive to than the direct-detection
experiments. For dark matter annihilation to heavy-quarks, the projected sensitivity of DUNE is
weaker than current (future) Super (Hyper) Kamiokande experiments. However, for the light-quark
channel, only the spike is observable and DUNE will be the most sensitive experiment.