According to the tidal stirring model, late type, rotationally supported dwarfs resembling present day dwarf irregular (dIrr) galaxies can transform into dwarf spheroidals (dSphs) via interactions with Milky-Way-sized hosts. We perform collisionless N-body simulations to investigate for the first time how tidal stirring depends on the dark matter (DM) density distribution in the central stellar region of the progenitor disky dwarf. Specifically, we explore various asymptotic inner slopes γ of the dwarf DM density profiles (ρvpropr -γ). For a given orbit inside the primary galaxy, rotationally supported dwarfs embedded in DM halos with core-like distributions (γ = 0.2) and mild density cusps (γ = 0.6) demonstrate a substantially enhanced likelihood and efficiency of transformation into dSphs compared to their counterparts with steeper DM density profiles (γ = 1). Such shallow DM distributions are akin to those of observed dIrrs highlighting tidal stirring as a plausible model for the Local Group (LG) morphology-density relation. When γ < 1, a single pericentric passage can induce dSph formation and disky dwarfs on low-eccentricity or large-pericenter orbits are able to transform; these new results allow tidal stirring to explain virtually all known dSphs across a wide range of distances from their hosts. A subset of disky dwarfs initially embedded in DM halos with shallow density profiles are eventually disrupted by the primary; those that survive as dSphs are generally on orbits with lower eccentricities and/or larger pericenters compared to those of typical cold dark matter satellites. The latter could explain the peculiar orbits of several LG dSphs such as Fornax, Leo I, Tucana, and Cetus.