algorithm2e fully justified input / output blocks

Try this:

\documentclass{article}
\usepackage[algoruled, linesnumbered]{algorithm2e}
\usepackage{ragged2e}
\usepackage{xpatch}

\makeatletter
\xpatchcmd\SetKwInOut
  {\hangafter=1\parbox[t]}
  {\hangafter=1\justify\parbox[t]}
  {}{\fail}
\makeatother

\SetKwInOut{Input}{Input}
\DontPrintSemicolon

\begin{document}
\begin{algorithm}
    \caption{Getting things done}
    \Input{Many things with longish names that make the ragged right look ugly in my opinion. Many things with longish names that make the ragged right look ugly in my opinion. Many things with longish names that make the ragged right look ugly in my opinion.}
    \For{$k=1$ \KwTo $N$}{
        \tcp{Do something}
        $x_k^- = A_{k-1} x_{k-1}^+ + B_{k-1} u_{k-1}$ \;
        $P_{k}^- = A_{k-1} P_{k-1}^+ A_{k-1}^T + Q_{k-1}$ \;
        \tcp{And now do something else}
        $K_k = P_{k}^- C_k^T (C_k P_{k}^- C_k^T + R_{k})^{-1}$ \;
        $x_{k}^+ = x_{k}^- + K_{k}(y_{k}-C_k x_{k}^-)$ \;
        $P_{k}^+ = (I - K_{k} C_k) P_{k}^-$ \;
    }
\end{algorithm}
\end{document}

Some explanations:

When calling \SetKwInOut{Input}{Input}, it defines \Input by

% here #1 is first arg of \SetKwInOut, here it is "Input"
\algocf@newcommand{#1}[1]{%
    \ifthenelse{\boolean{algocf@hanginginout}}{\relax}{\algocf@seteveryparhanging{\relax}}%
    \ifthenelse{\boolean{algocf@inoutnumbered}}{\relax}{\algocf@seteveryparnl{\relax}}%
    % The following line control the output of \Input{##1}.
    % This paragraph starts at \parbox, hence the change of paragraph
    % indent (\justify) should be inserted before \parbox.
    {\let\\\algocf@newinout\hangindent=\inoutindent\hangafter=1\parbox[t]{\inoutsize}{\KwSty{#2\algocf@typo\hfill:}}~##1\par}%
    \algocf@linesnumbered% reset the numbering of the lines
    \ifthenelse{\boolean{algocf@hanginginout}}{\relax}{\algocf@reseteveryparhanging}%
}