11. (10 points) If the equation with respect to $x$
$$
\frac{2 k}{x-1}-\frac{x}{x^{2}-x}=\frac{k x+1}{x}
$$

has only one solution, find the value of $k$.
When $k=0$, equation (1) has a unique solution $x=\frac{1}{2}$.
When $k \neq 0$, the discriminant of equation (1) is positive,

but the original equation has only one root, so one of the roots must be an extraneous root of the original equation. The extraneous root can only be 0 or 1. Clearly, 0 is not a root of (1), so $x=1$ is a root of equation (1). Substituting $x=1$ into equation (1) gives $k=\frac{1}{2}$. In this case, the root of the original equation is $-\frac{1}{k}=-2$. Therefore, $k=0$ or $\frac{1}{2}$.