Prove the languages |L| = 2 and |L| $\not=$ 2 to be non-Turing recognizable or non-recursively enumerable
If I can show that $A_{TM}$ is mapping reducible to $L_{=2}$, ...


That is a nice idea. Given $\langle M, w \rangle$, can you construct a TM $M_1$ that accepts only one word, the word $w$ if and only if $M$ accepts $w$? 

Then, can you change $M_1$ so that it accepts another word that is different from $w$?

You can also show that $A_{TM}$ is mapping reducible to $L_{\not=2}$ by creating a TM $M_3$ which accepts 2 words if $M$ does not accepts $w$ and 3 words otherwise.