The IceCube Neutrino Observatory, which instruments 1$\,$km$^3$ of clear ice at the geographic South Pole, was mainly designed to detect particles with energies in the multi-GeV to PeV range. Due to ice temperatures between $-20^\circ$C to $-43^\circ$C and the low radioactivity of the ice, the dark noise rates of the 5160 photomultiplier tubes forming the IceCube lattice are of order 500 Hz, which is particularly low for 10 inch photomultipliers. Therefore, IceCube can extend its searches to bursts of $\mathcal{O}$(10\,MeV) neutrinos lasting several seconds, which are expected to be produced by Galactic core collapse supernovae. By observing a uniform rise in all photomultiplier rates, IceCube can provide a particularly high statistical precision for the neutrino rate from supernovae in the inner part of our Galaxy ($<$ 20 kpc). In this paper, the tools and the method to study potential obscured or failed core collapse supernovae in our Galaxy are presented. The analysis will be based on 3911 days of IceCube data taken between April 17, 2008 and December 31, 2018.