Solution:

Cross multiplying gives $(x - 1)(x - 6) = (x - 2)(x - k),$ which expands to $$x^2 - 7x + 6 = x^2 - (2 + k)x + 2k.$$

Cancelling $x^2$ leaves $(k - 5)x = 2k - 6,$ so $x = \dfrac{2k - 6}{k - 5}.$ This has no solution exactly when $k = 5,$ since then the coefficient of $x$ is $0$ while the right side is nonzero.

Thus, the correct answer is E.