prove that there is a complete language in $L \cup \{A_{TM}\}$
We will show that $A_{TM}$ is complete for your class.

In order to show that, we need to show that for every language $A\in L\cup \{A_{TM}\}$, it holds that $A\le_L A_{TM}$.

First, if $A=A_{TM}$, then the trivial reduction suffices. That is, a reduction that given input $x$, return $x$. Clearly $x\in A_{TM}\iff x\in A_{TM}$.

Now, if $A\in L$, we need to work slightly harder. We need to show that $A\le_L A_{TM}$. In order to do that, we construct a reduction from $A$ to $A_{TM}$, as follows: let $M$ be a deterministic TM that decides $A$ in logspace (why does there exist such a machine?).

Now, the reduction works as follows: given input $x$, the reduction uses $M$ to check whether $x\in A$. If $x\in A$, the reduction outputs $\langle\epsilon, T_1\rangle$, where $T_1$ is a TM that accepts every input immediately. If $x\notin A$ the reduction outputs $\langle\epsilon, T_2\rangle$, where $T_2$ is a TM that rejects every input immediately. Note that $T_1,T_2$ are fixed, i.e. they are hard-coded in the reduction.

Try to prove that this reduction is correct, and that it is indeed in logspace.