Answer: D

Solution:

To find the number of digits, we can try to express this number in terms of powers of $10.$

We get that $$\begin{align*} 4^5 \cdot 5^{10} &= 2^{10} \cdot 5^{10} \\ &= 10^{10}. \end{align*}$$

This shows that the desired number is $1$ followed by $10$ zeros, for a total of $11$ digits.

Thus, D is the correct answer.