Reducing a problem to Halt
First of all I find it rather artificial to argue with reductions, since a more direct argument is applicable here. However, you can of course do it.

I think your approach follows basically the right direction. But it is not a clean reduction. Here is how I would phrase it.

We want to show that ${\sf 2Strings}$ is recognizable by showing ${\sf 2Strings}\le_m {\sf Halt}$. The reduction  goes as follows: Assume we have  a TM $M$ and based om $M$ we define a different TM $M'$. Let us first define a NTM $N$:

 0. Delete the input
 1. Guess two words u and w
 2. If u=w cycle
 3. Simulate u on M
 4. Simulate w on M
 5. Accept if both simulation accepted, otherwise cycle


Now let $M'$ be the deterministic version of $N$. The reduction maps $\langle M \rangle $ to $\langle M',\varepsilon \rangle$. By construction, 
$$
\begin{align}
\langle M \rangle \in {\sf 2String} &amp; \iff N \text{ accepts every input}\\
 &amp; \iff M' \text{stops on every input}\\
 &amp; \iff M' \text{stops on }\varepsilon \\
 &amp; \iff \langle M',\varepsilon\rangle \in {\sf Halt} \\
\end{align}
$$