I'll start by describing the notation that I'll use.

I'll think of elements of $\mathbb{Z}^\mathbb{N}$ as infinite column vectors
$${\bf x}=\begin{pmatrix}
x_0\\x_1\\x_2\\\vdots
\end{pmatrix}$$
of integers, with rows indexed by $\mathbb{N}$.

The $i$th "unit vector" will be denoted by ${\bf e(i)}$. I.e., $e(i)_i=1$ and $e(i)_j=0$ for $j\neq i$.

As described in the question, Specker's result implies that endomorphisms of $\mathbb{Z}^\mathbb{N}$ are of the form ${\bf x}\mapsto A{\bf x}$ where
$$A=\begin{pmatrix}
a_{00}&a_{01}&a_{02}&\cdots\\
a_{10}&a_{11}&a_{12}&\cdots\\
a_{20}&a_{21}&a_{22}&\cdots\\
\vdots&\vdots&\vdots&\ddots
\end{pmatrix}$$
is an infinite matrix of integers, with rows and columns indexed by $\mathbb{N}$, that is row-finite (i.e., each row has only finitely many non-zero entries).

Now suppose that $\alpha:\mathbb{Z}^\mathbb{N}\to\mathbb{Z}^\mathbb{N}$ is a surjective group homomorphism, given by a row-finite matrix $A$.

To prove that $\alpha$ is split, we need to find a right inverse. In fact,  it suffices to find another row-finite matrix $B$ such that $AB$ is lower unitriangular (i.e., diagonal entries are all $1$ and entries above the diagonal are all $0$), since such a matrix is invertible.

Fix $n\geq0$, and decompose $\mathbb{Z}^\mathbb{N}=G_n\oplus H_n$, where $G_n\cong\mathbb{Z}^n$ is the subgroup consisting of those ${\bf a}$ with $a_i=0$ for $i\geq n$, and $H_n\cong\mathbb{Z}^\mathbb{N}$ is the subgroup consisting of those ${\bf a}$ with $a_i=0$ for $i<n$.

Since $\alpha$ is surjective, $\mathbb{Z}^\mathbb{N}=\alpha\left(G_n\right)+\alpha\left(H_n\right)$.

Consider the quotient maps 
$$\mathbb{Z}^\mathbb{N}\stackrel{\theta}{\to} \mathbb{Z}^\mathbb{N}/\alpha\left(H_n\right)\stackrel{\varphi}{\to} \left(\mathbb{Z}^\mathbb{N}/\alpha\left(H_n\right)\right)/T\left(\mathbb{Z}^\mathbb{N}/\alpha\left(H_n\right)\right),$$ 
where $T(X)$ denotes the torsion subgroup of an abelian group $X$.

Since $\left(\mathbb{Z}^\mathbb{N}/\alpha\left(H_n\right)\right)/T\left(\mathbb{Z}^\mathbb{N}/\alpha\left(H_n\right)\right)$ is a finite rank free abelian group, using Specker's result again implies that $\varphi\theta({\bf e(i)})=0$, or equivalently $\theta({\bf e(i)})\in T\left(\mathbb{Z}^\mathbb{N}/\alpha\left(H_n\right)\right)$, for all but finitely many $i$. 

Since $T\left(\mathbb{Z}^\mathbb{N}/\alpha\left(H_n\right)\right)$ is finite, for all but finitely many $i$ with $\theta({\bf e(i)})\in T\left(\mathbb{Z}^\mathbb{N}/\alpha\left(H_n\right)\right)$, there is $i'>i$ with $\theta({\bf e(i')})=\theta({\bf e(i)})$, or equivalently ${\bf e(i)}-{\bf e(i')}\in\alpha(H_n)$.

So we can choose a sequence $0=t_0<t_1<t_2<\dots$ of integers such that for every $i\geq t_n$ there is some $i'>i$ and some ${\bf b(i)}\in H(n)$ with ${\bf e(i)}-{\bf e(i')}=\alpha\left({\bf b(i)}\right)$. For each $i$, do this with the largest $n$ such that $i\geq t_n$, and let $B$ be the matrix whose $i$th column is ${\bf b(i)}$.

Then $B$ is row-finite, and $AB$ is lower unitriangular.