Building on the striking similarity between the structure of the spindle during mitosis in living cells and nematic textures in confined liquid crystals, we use a continuum model of two-dimensional nematic liquid crystal droplets to examine the physical aspects of cell division. The model investigates the interplay between bulk elasticity of the microtubule assembly, described as a nematic liquid crystal, and surface elasticity of the cell cortex, modeled as a bounding flexible membrane, in controlling cell shape and division. The centrosomes at the spindle poles correspond to the cores of the topological defects required to accommodate nematic order in a closed geometry. We map out the progression of both healthy bipolar and faulty multi-polar division as a function of an effective parameter that incorporates active processes and controls centrosome separation. A robust prediction, independent of energetic considerations, is that the transition from a single cell to daughters cells occurs at critical value of this parameter. Our model additionally suggests that microtubule anchoring at the cell cortex may play an important role for successful bipolar division. This can be tested experimentally by regulating microtubule anchoring.