OP stated in the question that:
I've done a Google search regarding this question (How does $\ce{KI}$ catalyze the reaction with $\ce{H2O2}$?), and have found that there is no place online which answers this question to the degree that I want.
I think OP hasn't put enough time to find out that this experiment (Decomposition of hydrogen peroxide catalyzed by iodide) is a very popular experiment amongst first year laboratory programs around most of universities in United States and around the world. For example, see Ref.1, in which it states that $\ce{H2O2}$ can act either as a oxidizing or reducing reagent:
$$\ce{H2O2 + 2H+ + 2e- <=> 2H2O} \qquad \mathrm{E^\circ} = \pu{1.776 V} \tag1$$
$$\ce{O2 + 2H+ + 2e- <=> H2O2} \qquad \mathrm{E^\circ} = \pu{0.815 V} \tag2$$
The equations $(1)$ and $(21)$ shows that $\ce{H2O2}$ can undergo spontaneous self-oxidation-reduction, and is therefore, thermodynamically unstable. Accordingly, $\ce{H2O2}$ decomposes spontaneously and exothermically, but slowly to oxygen and water at room temperature. The slow reaction rate is due to its relatively large activation energy of $\pu{76 kJ mol-1}$ (Ref.2). However, it would decompose at a much faster rate when a catalyst is introduced. For example, the large activation energy is lowered to ~$\pu{41 kJ mol-1}$ when $\ce{MnO2}$ is used as the catalyst and proceeds with a heat of reaction of $\pu{−99 kJ mol-1}$ (Ref.2). The iodide-catalyzed decomposition reaction proceeds with a heat of reaction of $\pu{−98.3 kJ mol-1}$ (Ref.3).
According to Ref.1, $\ce{I-/IO-}$ redox couple with $\mathrm{E^\circ} = \pu{0.55 V}$ and $\ce{I2/I-}$ redox couple with $\mathrm{E^\circ} = \pu{0.535 V}$ are ideal systems to catalyze $\ce{H2O2}$ decomposition. If first couple is used, the rate determining step would be:
$$\ce{H2O2 + I- -> IO- + H2O} \tag3$$
The same the rate determining step is given in an older reference (Ref.4), so that the mechanism might be correct. The same group, Liebhafsky & Coworkers has mentioned in later publication (Ref.5) that:
The first successful quantitative investigation in chemical kinetics, completed in 1866 by Harcourt and Esson, proved that the rate of the reaction, $$\ce{H2O2 + 2H+ + 3I- -> 2H2O + I3-} \tag4$$ is proportional to the amounts of peroxide and iodide present, and that it is “accelerated" by the addition of acid. A quarter century later Magnanini in a remarkable investigation, which we shall subsequently consider in some detail, demonstrated how this “acceleration" is related to the acid added.
The authors have measured the rate of the reaction (given in the equation $(4)$) for the first time under the simplest experimental conditions, with the result that the law, previously established (by Magnanini):
$$\frac{d[\ce{I3-}]}{dt} = k_1^\circ \frac{[\ce{I-}]}{[\ce{H2O2}]} + k_1[\ce{H+}][\ce{I-}][\ce{H2O2}] \tag5$$
As a result, they have concluded that the simplest kinetic interpretation of this rate law assumes following two rate determining steps:
$$\ce{H2O2 + I- ->[k_1^\circ] H2O + IO-} \tag6$$
$$\ce{H2O2 + H+ + I- ->[k_1] H2O + HIO} \tag7$$
And, both steps proceed simultaneously and independently. At $\pu{25 ^\circ C}$, they have reported measured values of $k_1^\circ $ and $k_1$ as $0.69$ and and $10.5$, respectively.
References:
- John C. Hansen, "The Iodide-Catalyzed Decomposition of Hydrogen Peroxide: A Simple Computer-Interfaced Kinetics Experiment for General Chemistry," J. Chem. Educ. 1996, 73(8), 728-732 (ODI: https://doi.org/10.1021/ed073p728).
- William Sweeney, James Lee, Nauman Abid, and Stephen DeMeo, "Efficient Method for the Determination of the Activation Energy of the Iodide-Catalyzed Decomposition of Hydrogen Peroxide," J. Chem. Educ. 2014, 91(8), 1216–1219 (ODI: https://doi.org/10.1021/ed500116g).
- Tomoyuki Tatsuoka and Nobuyoshi Koga, "Energy Diagram for the Catalytic Decomposition of Hydrogen Peroxide," J. Chem. Educ. 2013, 90(5), 633−636 (ODI: https://doi.org/10.1021/ed400002t).
- Herman A. Liebhafsky, "The Catalytic Decomposition of Hydrogen Peroxide by the Iodine-Iodide Couple. II & III. The Rate of Oxidation in Neutral, and in Acid, Solution of Hydrogen Peroxide by Iodine," J. Am. Chem. Soc. 1932, 54(9), 3499–3508 (ODI: https://doi.org/10.1021/ja01348a003).
- Herman A. Liebhafsky and Ali Mohammad, "The Kinetics of the Reduction, in Acid Solution, of Hydrogen Peroxide by Iodide Ion," J. Am. Chem. Soc. 1933, 55(10), 3977–3986 (ODI: https://doi.org/10.1021/ja01337a010).