It is well known that in organic solids the collision of two excitons can give rise to delayed fluorescence (DF). Revived interest in this topic is stimulated by the current endeavor towards the development of efficient organic optoelectronic devices such as organic light-emitting diodes (OLEDs) and solar cells, or sensitizers used in photodynamic therapy. In such devices, triplet excitations are ubiquitously present but their annihilations can be either detrimental, e.g., giving rise to a roll-off of intensity in an OLED, or mandatory, e.g., if the sensitizer relies on up-conversion of long-lived low-energy triplet excitations. Since the employed materials are usually noncrystalline, optical excitations migrate via incoherent hopping. Here, we employ kinetic Monte Carlo simulations (KMC) to study the complex interplay of triplet-triplet annihilation (TTA) and quenching of the triplet excitations by impurities in a single-component system featuring a Gaussian energy landscape and variable system parameters such as the length of the hopping sites, i.e., a conjugated oligomer, the morphology of the system, the degree of disorder (σ), the concentration of triplet excitations, and temperature. We also explore the effect of polaronic contributions to the hopping rates. A key conclusion is that the DF features a maximum at a temperature that scales with σ/kBT. This is related to disorder-induced filamentary currents and thus locally enhanced triplet densities. We predict that a maximum for the TTA process near room temperature or above requires typically a disorder parameter of at least 70 meV.