Take $v$ a smooth incompressible vector field and call $w$ its vorticity.


If $w\in L^1(\mathbb{R}^2)$, the number
\begin{align*}
\alpha := \int_{\mathbb{R}^2} w \,dx,
\end{align*}
is well-defined. 

First fix $\psi\in\mathscr{D}(\mathbb{R}_+^*)$ (i.e. smooth and compactly supported) such as 
\begin{align*}
\int_0^\infty \psi(r)\,r\,dr = \alpha.
\end{align*}

Since $\psi$ vanishes totally in $0$ (in particular $\psi'(0)=0$ which is what we need here), you have the existence of a smooth function $\overline{w}$ such as $\psi'(x)=x \overline{w}(x)$. 

Define the radially symmetric vector field $b(x):=\psi(\|x\|)$, so that 
\begin{align*}
\int_{\mathbb{R}^2}b\, dx = \alpha.
\end{align*}

Remark that you have also 
\begin{align*}
b(x) = \int_0^{\|x\|} \overline{w}(s)\,s \,ds.
\end{align*}
Now, by construction the vector field $u:=v-K_2* b$ satisfies
\begin{align*}
\text{div}\,u &= 0,\\
\int_{\mathbb{R}^2} \text{curl}\,u &= 0,
\end{align*}
so that thanks to the Taylor expansion you mentionned, one has $u\in L^2(\mathbb{R}^2)$.