  In this paper, we study an impact of Leibniz algebras on the algebraic
structure of $\mathbb{N}$-graded vertex algebras. We provide easy ways to
characterize indecomposable non-simple $\mathbb{N}$-graded vertex algebras
$\oplus_{n=0}^{\infty}V_{(n)}$ such that $\dim V_{(0)}\geq 2$. Also, we examine
the algebraic structure of $\mathbb{N}$-graded vertex algebras
$V=\oplus_{n=0}^{\infty}V_{(n)}$ such that $\dim~V_{(0)}\geq 2$ and $V_{(1)}$
is a (semi)simple Leibniz algebra that has $sl_2$ as its Levi factor. We show
that under suitable conditions this type of vertex algebra is indecomposable
but not simple. Along the way we classify vertex algebroids associated with
(semi)simple Leibniz algebras that have $sl_2$ as their Levi factor.
