Note that $10^{2021}$ has 2021 zeroes, with a one as the first digit. After adding 2021, it is a $1$ followed by 2017 zeroes, ending with a 2021.
Although the exact locations of the digits of $2021^2$ will vary, The digits will remain identical, which is the $1$ on the first digit, with $442484441$ spread out. That is equal to a total of $\boxed{36}$