Is $K' = \{ w \in \{0,1\}^* | M_w$ Halts on $w \}$, where $M_w$ is the TM whose encoding is $w$, equivalent to the halting problem?
The two languages aren't equal. However both $K$ and $K'$ are undecidable. 

Proof:

We know that:
$K = \{ \langle M , v \rangle | M$ halts on $v$ }. Let us construct another language: $K'' = \{ \langle M , \langle M\rangle \rangle | M$ halts on $\langle M \rangle \}$. Note that $K'' \subset K$ because we replaced $v$ by $\langle M \rangle$ (encoding of $M$). Also note the similarity of $K'$ and  $K''$.

If you prove that $K'$ is undecidable, we can conclude that $K$ is undecidable.

You might have proved that $K'$ is undecidable.