This is to extend Christian Remling's [comment](https://mathoverflow.net/questions/432168/fix-positive-t-construct-a-n-in-mathbb-rn-such-that-inf-x-x-a-n-2#comment1112378_432168) to all real $t>0$, with an explicit lower bound on $K/M$, where $K:=K_n(a,t)$ and $M:=M_n(a,t)$. 

$\newcommand\norm[1]{\lVert#1\rVert}$The key here, as in Christian Remling's [comment](https://mathoverflow.net/questions/432168/fix-positive-t-construct-a-n-in-mathbb-rn-such-that-inf-x-x-a-n-2#comment1112378_432168), is the observation that $\norm x_1\ge\norm x_2$ for all $x\in\mathbb R^n$. Indeed, this observation implies 
$$K
\ge\inf_{x\in\mathbb R^n}\bigl(\lvert\norm x_2-\norm a_2\rvert+t\norm x_2\bigr)
=\inf_{u\ge0}\bigl(\lvert u-\norm a_2|+tu\bigr)
=\min(1,t)\norm a_2$$
and 
$$M\le\norm a_2,$$
whence 
$$\frac KM\ge\min(1,t).$$
(In particular, $K\ge M$ if $t\ge1$.)