We note $A$ the annuity, so that $V^{swap} = A(s - K)$ so that $\frac{\partial V^{swap}}{\partial s} = A$. As the chain rule gives $$\frac{\partial V^{swap}}{\partial r} = \frac{\partial s}{\partial r} \frac{\partial V^{swap}}{\partial s}$$ we get that $$\frac{\partial s}{\partial r} = \frac{1}{A} \frac{\partial V^{swap}}{\partial r}$$ and as the chain rule gives also : $$\frac{\partial V^{swaption}}{\partial r} = \frac{\partial s}{\partial r} \frac{\partial V^{swaption}}{\partial s}$$ which combined with the previous equation gives $$\frac{\partial V^{swaption}}{\partial r} = \frac{1}{A} \frac{\partial V^{swap}}{\partial r} \frac{\partial V^{swaption}}{\partial s}$$ and diving both sides by $\frac{\partial V^{swap}}{\partial r}$ gives $$L = \frac{1}{A} \frac{\partial V^{swaption}}{\partial s}$$

We note $A$ the annuity, so that $V^{swap} = A(s - K)$ so that $\frac{\partial V^{swap}}{\partial s} = A$. As the chain rule gives $$\frac{\partial V^{swap}}{\partial r} = \frac{\partial s}{\partial r} \frac{\partial V^{swap}}{\partial s}$$ we get that $$\frac{\partial s}{\partial r} = \frac{1}{A} \frac{\partial V^{swap}}{\partial r}$$ and as the chain rule gives also : $$\frac{\partial V^{swaption}}{\partial r} = \frac{\partial s}{\partial r} \frac{\partial V^{swaption}}{\partial s}$$ which combined with the previous equation gives $$\frac{\partial V^{swaption}}{\partial r} = \frac{1}{A} \frac{\partial V^{swap}}{\partial r} \frac{\partial V^{swaption}}{\partial s}$$ and diving both sides by $\frac{\partial V^{swap}}{\partial r}$ gives $$L = \frac{1}{A} \frac{\partial V^{swaption}}{\partial s}$$ where $A$ is given by the spot discount curve and $\frac{\partial V^{swaption}}{\partial s}$ by the new model.