Final answer:
The centrosomes are the structures responsible for organizing spindle fibers and ensuring proper chromosome separation during mitosis by forming the mitotic spindle.
Step-by-step explanation:
The structures that contribute to organizing spindle fibers, separating chromatids, and distributing chromosomes during mitosis are the centrosomes. The centrosome is a major cellular structure responsible for the growth of microtubules and includes a pair of centrioles. As mitosis progresses, the centrosomes migrate to opposite ends of the cell, and spindle fibers form from microtubules that extend out from the centrosomes. These spindle fibers then attach to the chromosomes via their centromeres and play a vital role in aligning chromosomes during metaphase and separating sister chromatids during anaphase.