I don't know why these two method yield different solutions for marginal product of labour
I am not sure why you expect method two to work, as generally
$$
\frac{\partial F(K,N)}{\partial N} \neq \frac{\partial F(K,N)}{\partial K}\frac{\partial K}{\partial N}.
$$
These are partial derivatives. Unless information to the contrary exists
$$\frac{\partial K}{\partial N} = 0.$$
You can easily see that these formulas are not connected by calculating the area of a rectangle which sides have lengths of $a,b$. Thus $A(a,b) = a\cdot b$ and
$$
\frac{\partial A(a,b)}{\partial b} = a \neq b \cdot 0 = \frac{\partial A(a,b)}{\partial a}\frac{\partial a}{\partial b}.
$$
In case you insist on homogeneity of degree one you can take the square root of the area.