Finding the inverse of the arc length function

Nice idea!

As long as the function $g(x)$ is well-behaved, we have the following very important result. Let $$G(x)=\int_c^x g(t)dt$$. Then $G'(x)=g(x)$.

This result (and some related ones) is called the Fundamental Theorem of (Integral) Calculus.

Now let us apply that to your problem. We obtain $$L'(x)=\sqrt{1+(f'(x))^2}$$

Use the above equation to solve for $f'(x)$ in terms of $L'(x)$. If you take $L(x)$ as known, you have found an explicit formula for $f'(x)$, and all you need to do is to integrate.

Now comes the unfortunate part. For most pleasant functions $L(x)$, the resulting integration problem will be either difficult or more often impossible (in terms of standard functions).

I hope that this gives you something to play with. You will find out why there is such a limited number of different arclength problems in calculus books!

You can do it by a differential equation without getting $L(x)$ explicitly. If $\frac{dL}{dx} = \sqrt{1 + f'(x)^2}$, then $\frac{dx}{dL} = \frac{1}{\sqrt{1 + f'(x)^2}}$. Numerical methods can be used to solve this differential equation.