The main drawback with this approach is that it puts the entire speech into a BF buffer. The showframe is just to check alignment.
I played around with fancyhdr and \pagestyule{myheadings) but finally went with \afterpage.
\documentclass[paper=a5]{article}
\usepackage{blindtext}
\usepackage{afterpage}
\usepackage{showframe}
\newsavebox{\BFbuffer}
\newsavebox{\headbox}
\newenvironment{drone}{\global\setbox\headbox=\hbox{Test}%
\setbox\BFbuffer=\vbox\bgroup}%
{\egroup\newpage
\loop\ifdim\ht\BFbuffer>0pt
\setbox0=\vsplit\BFbuffer to \textheight
\ifdim\ht\BFbuffer>0pt
\setbox1=\copy0
\setbox2=\vsplit1 to \dimexpr \ht1-\baselineskip\relax
\global\setbox\headbox=\hbox{\usebox1}%
\afterpage{\rlap{\raisebox{\dimexpr \topskip+\headsep}[0pt]{\usebox\headbox}}\vskip-\baselineskip}%
\fi
\unvbox0\pagebreak
\repeat}
\begin{document}
\begin{drone}
\blindtext
\blindtext
\blindtext
\blindtext
\blindtext
\blindtext
\blindtext
\blindtext
\end{drone}
\end{document}