Show that $A_\mathrm{LBA}$ is PSPACE-Complete?
As I understand, you are asking two questions. Let me address them separately: 


You don't. Your assumption on $M$ did not require anything more than $L(M) = A$, which is too weak for such a proof. You need to use $A \in \textbf{PSPACE}$ and require the existence of a polynomial $p:  \mathbb{N} \to \mathbb{N}$ such that $M$ uses at most space $p(n)$ for any input of length $n$.Incidentally, you also need this to know what $k$ should be. In your proof, you give no explicit way of computing $k$ other than simulating $M'$, but you do not state why this could ever be done in poly-time. (In fact, this could only ever work for arbitrary $M$ if $\textbf{P} = \textbf{PSPACE}$.) 
The description of $M$ remains constant as the length of the input $x$ increases, so it is negligible, asymptotically speaking. 




The real issue

It appears your approach is based on fixing $A$ and $x$ and then constructing an LBA $M''$ to simulate $M$ on $x$. This is not how reductions work. $A$ and $M$ should be fixed, but you need to keep $x$ arbitrary, otherwise the proof is quite incorrect (since the result of running $M$ on $x$ could be simply hardcoded into a TM which produces the result immediately upon reading $x$ as input; this proves absolutely nothing).