Solution:

The square of the nine-digit repunit is the palindrome $$111{,}111{,}111^2 = 12{,}345{,}678{,}987{,}654{,}321.$$

Its digits are $1, 2, \ldots, 9, 8, \ldots, 1,$ so the sum is $$2(1 + 2 + \cdots + 8) + 9 = 2 \cdot 36 + 9 = 81.$$

Thus, the correct answer is E.