Whilst many vertebrates appear externally left-right symmetric, the arrangement of internal organs is asymmetric. In zebrafish, the breaking of left-right symmetry is organised by Kupffer's Vesicle (KV): an approximately spherical, fluid-filled structure that begins to form in the embryo 10 hours post fertilisation. A crucial component of zebrafish symmetry breaking is the establishment of a cilia-driven fluid flow within KV. However, it is still unclear (a) how dorsal, ventral and equatorial cilia contribute to the global vortical flow, and (b) if this flow breaks left-right symmetry through mechanical transduction or morphogen transport. Fully answering these questions requires knowledge of the three-dimensional flow patterns within KV, which have not been quantified in previous work. In this study, we calculate and analyse the three-dimensional flow in KV. We consider flow from both individual and groups of cilia, and (a) find anticlockwise flow can arise purely from excess of cilia on the dorsal roof over the ventral floor, showing how this vortical flow is stabilised by dorsal tilt of equatorial cilia, and (b) show that anterior clustering of dorsal cilia leads to around 40 % faster flow in the anterior over the posterior corner. We argue that these flow features are supportive of symmetry breaking through mechano-sensory cilia, and suggest a novel experiment to test this hypothesis. From our new understanding of the flow, we propose a further experiment to reverse the flow within KV to potentially induce situs inversus.