You can add heading=subbibliography when issuing \printbibliography.
Also, load biblatex with the option defernumbers=true when printing different bibliographies with different types.
\documentclass{article}
\usepackage{lipsum}
\usepackage[defernumbers=true]{biblatex}
\addbibresource{NZSM446.bib}
\begin{document}
\section{Sections Available}
\subsection{Subsection}
\subsubsection{Subsubsection}
\section{Bibliography}
\nocite{*}
\printbibliography[title={Books},type=book,sorting=nyt,heading=subbibliography]
\printbibliography[title={Articles},type=article,sorting=nyt,heading=subbibliography]
\end{document}

In response to your edited question, you can also add heading=subbibnumbered when issuing \printbibliography to achieve what you want.
Notice that, if you want, you can also substitute (for consistency)
\section{Bibliography}
with
\printbibheading[title={Bibliography},heading=bibnumbered]
So, the definitive MWE could be:
\documentclass{article}
\usepackage{lipsum}
\usepackage[defernumbers=true]{biblatex}
\addbibresource{NZSM446.bib}
\begin{document}
\section{Sections Available}
\subsection{Subsection}
\subsubsection{Subsubsection}
\nocite{*}
\printbibheading[title={Bibliography},heading=bibnumbered]
\printbibliography[title={Books},type=book,sorting=nyt,heading=subbibnumbered]
\printbibliography[title={Articles},type=article,sorting=nyt,heading=subbibnumbered]
\end{document}
