You can manually break the algo at a suitable point with \algstore{bkbreak}...\algrestore{bkbreak}:
\documentclass{article}
\usepackage{algorithm}
\usepackage{algorithmicx}
\begin{document}
\begin{algorithm}
\begin{algorithmic}[1]
\State {em Lorem}
(...)
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\algstore{bkbreak}
\end{algorithmic}
\end{algorithm}
\begin{algorithm}
\begin{algorithmic}[1]
\algrestore{bkbreak}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\end{algorithmic}
\end{algorithm}
\end{document}
For automatic page breaks, don't wrap it in a floating environment:
\documentclass{article}
\usepackage{algorithm}
\usepackage{algorithmicx}
\begin{document}
%\begin{algorithm}
\begin{algorithmic}[1]
\State {em Lorem}
(...)
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\State {Final}
\end{algorithmic}
%\end{algorithm}
\end{document}