Expected value of a random number of ones and nines

An $8$-digit number $X$ is generated in two steps. First draw $k$ uniformly from $\{0,1,\dots,8\}$. Then choose uniformly among all $8$-digit strings containing exactly $k$ ones, with every other digit a nine. What is $\mathbb{E}[X]$?

55555555. Swapping every $1$ with a $9$ maps strings with $k$ ones to strings with $8-k$ ones, and $k$ and $8-k$ are equally likely, so each digit is $1$ or $9$ with probability $\tfrac12$. Each digit therefore averages $5$, and by linearity $\mathbb{E}[X]=5\times11111111$.