We consider the triangular kagome XXZ-Ising model (TKL XXZ-Ising model) formed by inserting small triangles ("a-trimers") with XXZ spin-1 inside the triangles of the kagome lattice ("b-trimers"). It is a mixed spin system and can be solved exactly by transforming into the kagome lattice with the general transformation method for decorated spin systems. In the absence of an external field, we integrate out the quantum spins of the a-trimers and map the TKL model to the kagome Ising model exactly. We obtain the full phase diagram and their zero-temperature entropies (e.g., s_{max}=5.48895 per unit cell is given for the phase with the maximum entropy). When an external field is applied, 20 phases are found due to the quantum fluctuations of a-trimers. Moreover, the high spins in the a-trimers can lead to a stable quantized growth of the magnetization process in the Heisenberg limit.