To verify the solution, just apply the Laplace operator to it and see that it gives 0 indeed. Note that your solution is separable $\displaystyle u(x,y) = X(x)Y(y)$, which should ease the procedure.
No need for Laplace transforms. This is the Laplace equation.
You fixed $\displaystyle a$ and $\displaystyle b$ such that the boundary conditions are satisfied. Good. But you need to check that the solution satisfies the Laplace equation, i.e. $\displaystyle \nabla^2 u = {\partial^2 u \over \partial x^2} + {\partial^2 u \over \partial y^2} = 0$.
On the first page you checked that the Dirichlet boundary conditions $\displaystyle u(x,0) = 0$ and $\displaystyle u(x,L_y ) = 0$. This shows that 2 boundary conditions are satisfied. You need to compute the Laplacian of the function to show that the equation is true everywhere in the domain.