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 $$K\ge\int_{x\in\mathbb R^n}|\|x\|_2-\|a\|_2|+t\|x\|_2\ge\min(1,t)\|a\|_2$$ 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 $t=1$.)