Cancer with all its more than 200 variants continues to be a major health problem around the world with nearly 10 million deaths recorded in 2020, and leukemia accounted for more than 300,000 cases according to the Global Cancer Observatory. Although new treatment strategies are currently being developed in several ongoing clinical trials, the high complexity of cancer evolution and its survival mechanisms remain as an open problem that needs to be addressed to further enhanced the application of therapies. In this work, we aim to explore cancer growth, particularly chronic lymphocytic leukemia, under the combined application of CAR-T cells and chlorambucil as a nonlinear dynamical system in the form of first-order Ordinary Differential Equations. Therefore, by means of nonlinear theories, sufficient conditions are established for the eradication of leukemia cells, as well as necessary conditions for the long-term persistence of both CAR-T and cancer cells. Persistence conditions are important in treatment protocol design as these provide a threshold below which the dose will not be enough to produce a cytotoxic effect in the tumour population. In silico experimentations allowed us to design therapy administration protocols to ensure the complete eradication of leukemia cells in the system under study when considering only the infusion of CAR-T cells and for the combined application of chemoimmunotherapy. All results are illustrated through numerical simulations. Further, equations to estimate cytotoxicity of chlorambucil and CAR-T cells to leukemia cancer cells were formulated and thoroughly discussed with a 95% confidence interval for the parameters involved in each formula.