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It has been proposed that gravothermal collapse due to dark matter self-interactions (i.e. self-interacting dark matter, SIDM) can explain the observed diversity of the Milky Way (MW) satellites' central dynamical masses. We investigate the process behind this hypothesis using an $N$-body simulation of a MW-analogue halo with velocity dependent self-interacting dark matter (vdSIDM) in which the low velocity self-scattering cross-section, $σ_{T}/m_{x}$, reaches 100 cm$^{2}$g$^{-1}$; we dub this model the vd100 model. We compare the results of this simulation to simulations of the same halo that employ different dark models, including cold dark matter (CDM) and other, less extreme SIDM models. The masses of the vd100 haloes are very similar to their CDM counterparts, but the values of their maximum circular velocities, $V_{max}$, are significantly higher. We determine that these high $V_{max}$ subhaloes were objects in the mass range [$5\times10^{6}$, $1\times10^{8}$] $M_\odot$ at $z=1$ that undergo gravothermal core collapse. These collapsed haloes have density profiles that are described by single power laws down to the resolution limit of the simulation, and the inner slope of this density profile is approximately $-3$. Resolving the ever decreasing collapsed region is challenging, and tailored simulations will be required to model the runaway instability accurately at scales $<1$ kpc.