How can I use the Lagrange Inversion Theorem?

Question

I have a function $f(x)= x(\ln(x\ln x))$ and I want to use the Lagrange Inversion theorem to find its inverse $g(x)$ centered around a point $a$. The formula states that: $$g(x)=a+\sum_{n=1}^{\infty} \left(\lim_{x\to a}\frac{d^{n-1}}{dx^{n-1}}\left(\frac{x-a}{f(x)-f(a)}\right)^n \frac{(x-f(a))^n}{n!}\right).$$ I don't really know how to evaluate the limit. I have tried taking the limit before the taking the derivative, which is $$\lim_{x\to e}\left(\frac{x-e}{x(\ln(x\ln x))-e}\right)^n$$ which evaluates to $1/3^n$. I have tried taking the derivative first, for the first two values of $n$ I got $\frac {-1}{18e}$ and $\frac {-1}{27e}$ After this it becomes too slow to compute the third derivative manually or even in Wolfram Alpha, continuing the pattern I would say it would be $\frac {-1}{36e}$ but I truly have no idea. None of these alternatives seem to give me what I am looking for, what am I doing wrong?