In this work, we focus on modeling the flow of wormlike micellar solutions in a cross-slot device. In particular, we investigate the role that the breakage of wormlike micelles plays in the formation of lip vortices and the development of an asymmetric elastic instability. The cross-slot induces a planar elongational flow field around the stagnation point in the center of the geometry, and a shear-dominated flow in the inlet and outlet arms. The extensional flow can induce the emergence of a bifurcation, in which the symmetric flow undergoes an instability and becomes asymmetric at sufficiently large flow rates. We further show that this instability can be completely suppressed by creating micelles that are easier to break under imposed stretching. Furthermore, at larger flow rates, unlike comparable simulations of polymer solutions, the asymmetric flow remains steady in agreement with experiments. In addition, the shear flow can induce the formation of a recirculation region in the inlet arm along the channel wall directly upstream of the corner. In particular, we reveal the possibility of controlling the vortex regions via flow-induced breakage; as the chains become easier to break, the degree of shear thinning increases and the size of the vortices decreases. Ultimately, our predictions show that chain scission plays an important role in altering fluid flow.