Z-transform of an impulse signal in discrete time
There's a ROC (region of convergence) problem in your derivation.

Consider a decomposition of the signal as $x[n] = x_1[n] + x_2[n]$. Then in order for the $Z$ transform of $x[n]$ to exist, the corresponding $Z$-transforms of both $x_1[n]$ and $x_2[n]$ must exist for a set of $z$ which is the intersection of the individual ROCs of the components $x_1[n]$ and $x_2[n]$. If the intersection set is empty then the Z transform does not exist. Applying this to your signal yields.

$$x[n] = \sum_{k=-\infty}^{\infty} \delta[n-k] = u[n] + u[-n-1] = 1 ~~~~,~~~\text{for all n}$$

Where the individual $Z$-transform and their corresponding ROCs are
$$ u[n] \longleftrightarrow \frac{1}{1-z^{-1}} ~~~,~~~ ROC_1 ~~~ |z| &gt; 1 $$
$$ u[-n-1] \longleftrightarrow \frac{-1}{1-z^{-1}} ~~~,~~~ ROC_2 ~~~ |z| &lt; 1 $$

As you can see the intersection of $ROC_1$ and $ROC_2$ is empty; i.e; there is no set of values $z$ for which both of the components has convergent $Z$ transforms. Hence we conclude that the $Z$ transform $X(z)$ of the signal $x[n]=1$ for all $n$ does not exist.