ML estimation - solve for x
It looks okay to me. If you define empirical mean $\hat{\mu}_d = \frac 1N \sum \tilde{d}[n]$ and empirical second moment $\hat{\gamma}_d = \frac 1N \sum \tilde{d}^2[n]$ then you effectively have an equation of the form $$-N/x + A/x^3 + B/x^2 \stackrel{!}{=} 0,$$ where $A$ and $B$ directly depend on $\hat{\mu}_d, \hat{\gamma}_d, \mu_A, \sigma_A$. If you multiply this equation with $x^3$ it becomes a quadratic equation in $x$, which you can solve for $x$...
I get something like $$x^2 - \frac{\hat{\mu}_d \mu_A}{\sigma_A^2} x + \frac{\gamma_d}{\sigma_A^2} = 0$$ but that was scribbled down in a rush so I would have to check it again later. Can you confirm or correct?