how to mapping reduce any r.e. language to the diagonal language K?
For any r.e. language $L$, it is the domain of some M_l.

Consider the machine $M_n(x,y)=M_l(x)$, and now let $f(x)=s_1^1(n,x)$ (obtained from snm theorem). Hence for any $y$, $M_{f(x)}(y)=M_l(x)$

Then $L$ can be reduced to $K$ by $f$ :

if $x\in L$ iff M_l(x) halts iff $M_{f(x)}(f(x))$ halts iff $f(x)\in K$.