(i) Given \(\frac{dy}{dx} = 6x^2 + \frac{k}{x^3}\) and the gradient at \(P(1, 9)\) is 2, substitute \(x = 1\) into the derivative:

\(6(1)^2 + \frac{k}{(1)^3} = 2\)

\(6 + k = 2\)

Solving for \(k\), we get \(k = -4\).

(ii) To find the equation of the curve, integrate \(\frac{dy}{dx} = 6x^2 - \frac{4}{x^3}\):

\(y = \int (6x^2 - \frac{4}{x^3}) \, dx\)

\(y = 2x^3 + 2x^{-2} + c\)

Using the point \(P(1, 9)\), substitute \(x = 1\) and \(y = 9\):

\(9 = 2(1)^3 + 2(1)^{-2} + c\)

\(9 = 2 + 2 + c\)

\(c = 5\)

Thus, the equation of the curve is \(y = 2x^3 + 2x^{-2} + 5\).