Indirect detection strategies of Dark Matter (DM) entail searching for signals of DM annihilation or decay, typically in the form of excess positrons or high-energy photons above the astrophysical background, originating from (inferred) DM-rich environments. Due to their characteristics, dwarf spheroidal satellite galaxies (dSphs) of the Milky Way are considered very promising targets for indirect particle DM identification. To compare model predictions with the observed fluxes of product particles, most analyses of astrophysical data - which are generally performed via frequentist statistics - rely on estimating the abundance of DM by calculating the so-called J-factor. This quantity is usually inferred from the kinematic properties of the stellar population of a dSph, performing a Jeans analysis by means of Bayesian techniques. Previous works have, therefore, combined different statistical methods when analysing astrophysical data from dSphs. This thesis describes the development of a new, fully frequentist approach for constructing the profile likelihood curve for J-factors of dSphs, which can be implemented in indirect DM searches. This method improves upon previous ones by producing data-driven expressions of the likelihood of J, thereby allowing a statistically consistent treatment of the astroparticle and astrometric data from dSphs. Using kinematic data from twenty one satellites of the Milky Way, we derive estimates of their maximum likelihood J-factor and its confidence intervals. The analyses are performed in two different frameworks: the standard scenario of a collisionless DM candidate and the possibility of a self-interacting DM species. In the former case, the obtained J-factors and their uncertainties are consistent with previous, Bayesian-derived values. In the latter, we present prior-less estimates for the Sommerfeld enhanced J-factor of dSphs. In agreement with earlier studies, we find J to be overestimated by several orders of magnitude when DM is allowed is attractively self-interact. In both cases we provide the profile likelihood curves obtained. This technique is validated on a publicly available simulation suite, released by Gaia Challenge, by evaluating its coverage and bias. The results of these tests indicate that the method possesses good statistical properties. Lastly, we discuss the implications of these findings for DM searches, together with future improvements and extensions of this technique.