Is the following fuction computable?
You're using the terminology of recursion theory. Let me rephrase the argument in terms of computer programs.


$K$ is the set of all programs $x$ which halt when run on $x$ as input.
$K_1$ is the set of all programs $x$ which halt on some input.


Given a program $x$, consider the following program $f(x)$:


For $n=1,2,3,\ldots$: run program $x$ on inputs $1,\ldots,n$ for $n$ steps, and halt if $x$ halts of any of them halt.


The program $f(x)$ halts iff $x \in K_1$. In particular, $x \in K_1$ iff $f(x) \in K$. Since $f$ is computable, this gives a reduction from $K_1$ to $K$.