Something like this? Notice the options remember picture,overlay
for tikzpicture
(thanks to Paul Gaborit for reminding me of this):
\documentclass{article}
\usepackage{tikzpagenodes}
\usepackage{pgfornament}
\begin{document}
\begin{tikzpicture}[remember picture,overlay]
\draw ([yshift=-10pt]current page header area.south west) -- %
([xshift=-5pt,yshift=-10pt]current page header area.south) %
node[xshift=20pt] {\pgfornament[width=0.5cm]{1}};
\end{tikzpicture}
\end{document}

You had a spurious ;
before the node.
Perhaps is it something like this what you have in mind?
\documentclass{article}
\usepackage{tikzpagenodes}
\usepackage{pgfornament}
\begin{document}
\begin{tikzpicture}[remember picture,overlay]
\draw ([yshift=-10pt]current page header area.south west) --
([xshift=-15pt,yshift=-10pt]current page header area.south);
\draw ([xshift=15pt,yshift=-10pt]current page header area.south) --
([yshift=-10pt]current page header area.south east);
\node[yshift=-10pt] at (current page header area.south) {\pgfornament[width=0.5cm]{1}};
\end{tikzpicture}
\end{document}

Or this:
\documentclass{article}
\usepackage{tikzpagenodes}
\usepackage{pgfornament}
\begin{document}
\begin{tikzpicture}[remember picture,overlay]
\draw ([yshift=-10pt]current page header area.south west) --
node[fill=white,inner xsep=10pt] {\pgfornament[width=0.5cm]{1}}
([yshift=-10pt]current page header area.south east);
\end{tikzpicture}
\end{document}
The precaution mentioned in the answer to How to avoid empty space caused by TikZ might be useful too.