This is to extend Christian Remling's comment to all real $t>0$, with an explicit lower bound on $K/M$, where $K:=K_n(a,t)$ and $M:=M_n(a,t)$. 

The key here, as in Christian Remling's comment, is the observation that $\|x\|_1\ge\|x\|_2$ for all $x\in\mathbb R^n$. Indeed, this observation implies 
$$K
\ge\inf_{x\in\mathbb R^n}\big(|\|x\|_2-\|a\|_2|+t\|x\|_2\big)
=\inf_{u\ge0}\big(|u-\|a\|_2|+tu\big)
=\min(1,t)\|a\|_2$$
and 
$$M\le\max(1,t)\min(\|a\|_2,\|a\|_1)=\max(1,t)\|a\|_2,$$
whence 
$$\frac KM\ge\frac{\min(1,t)}{\max(1,t)}.$$
(In particular, $K\ge M$ if $t=1$.)