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Choice B is correct. Adding $9$ to each side of the given equation yields $x^2-2x=9$. To complete the square, adding $1$ to each side of this equation yields $x^2-2x+1=9+1$, or ${\left(x-1\right)}^2=10$. Taking the square root of each side of this equation yields $x-1=\pm \sqrt{10}$. Adding $1$ to each side of this equation yields $x=1\pm \sqrt{10}$. Since it's given that one of the solutions to the equation can be written as $1+\sqrt{k}$, the value of $k$ must be $10$.

Alternate approach: It's given that $1+\sqrt{k}$ is a solution to the given equation. It follows that $x=1+\sqrt{k}$. Substituting $1+\sqrt{k}$ for $x$ in the given equation yields ${\left(1+\sqrt{k}\right)}^2-2\left(1+\sqrt{k}\right)-9=0$, or $\left(1+\sqrt{k}\right)\left(1+\sqrt{k}\right)-2\left(1+\sqrt{k}\right)-9=0$. Expanding the products on the left-hand side of this equation yields $1+2\sqrt{k}+k-2-2\sqrt{k}-9=0$, or $k-10=0$. Adding $10$ to each side of this equation yields $k=10$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.