We report eclipse timing variation analyses of 26 compact hierarchical triple stars comprised of an eccentric eclipsing (inner) binary and a relatively close tertiary component found in the {em Kepler} field. We simultaneously fit the primary and secondary $O-C$ curves of each system for the light-travel time effect (LTTE), as well as dynamical perturbations caused by the tertiary on different timescales. For the first time, we include those contributions of three-body interactions which originate from the eccentric nature of the inner binary. These effects manifest themselves both on the period of the triple system, $P_2$, and on the longer apse-node timescale. We demonstrate that consideration of the dynamically forced rapid apsidal motion yields an efficient and independent tool for the determination of the binary orbits eccentricity and orientation, as well as the 3D configuration of the triple. Modeling the forced apsidal motion also helps to resolve the degeneracy between the shapes of the LTTE and the dynamical delay terms on the $P_2$ timescale, due to the strong dependence of the apsidal motion period on the triples mass ratio. This can lead to the independent determination of the binary and tertiary masses without the need for independent radial velocity measurements. Through the use of our analytic method for fitting $O-C$ curves we have obtained robust solutions for system parameters for the ten most ideal triples of our sample, and only somewhat less robust, but yet acceptable, fits for the remaining systems. Finally we study the results of our 26 system parameter fits via a set of distributions of various physically important parameters, including mutual inclination angle, and mass and period ratios.
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