1. **State the problem:** Simplify the expression $\frac{1}{5} + \left(\frac{2}{3} \times \frac{2}{3}\right) \div \frac{5}{8}$.\n\n2. **Recall the order of operations:** Multiplication and division come before addition. So first calculate $\frac{2}{3} \times \frac{2}{3}$, then divide by $\frac{5}{8}$, and finally add $\frac{1}{5}$.\n\n3. **Calculate the multiplication:** $$\frac{2}{3} \times \frac{2}{3} = \frac{2 \times 2}{3 \times 3} = \frac{4}{9}.$$\n\n4. **Divide by $\frac{5}{8}$:** $$\frac{4}{9} \div \frac{5}{8} = \frac{4}{9} \times \frac{8}{5} = \frac{4 \times 8}{9 \times 5} = \frac{32}{45}.$$\n\n5. **Add $\frac{1}{5}$:** $$\frac{1}{5} + \frac{32}{45} = \frac{9}{45} + \frac{32}{45} = \frac{41}{45}.$$\n\n6. **Final answer:** $$\boxed{\frac{41}{45}}.$$