sans serif math in tables and siunitx

If you specify the option detect-family for siunitx, the package would be able to detect that the units are used in sans-serif mode.

\usepackage[detect-family]{siunitx}

You could use the sansmath package for switching to sans-serif math:

\usepackage{sansmath}
\newcolumntype{k}{>{\sansmath$}l<{$\unsansmath}}

Use \sansmath and \unsansmath outside math mode, so with dcolumn before \DC@ and after \DC@end, respectively:

\newcolumntype{d}[1]{>{\sansmath\DC@{.}{ ,}{#1}}l<{\DC@end\unsansmath}}

With siunitx, there are a couple of potential approaches. First, as Stefan has said you can detect the current font:

\documentclass[paper=a4, fontsize=11pt]{scrartcl}
\usepackage{booktabs,lmodern,siunitx}
\usepackage{siunitx}

\newcolumntype{k}{%
>{$\mathsf\bgroup}l%
<{\egroup$}%
}


\begin{document}
\sffamily

\begin{center}
  \sisetup{table-format = 2.4,detect-family}
  \begin{tabular}{S
  k
  S[table-number-alignment = left]
  S[table-number-alignment = right]}
  \toprule
  {Some Values} & {Some Values} & {Some Values / \si{kg/m^2}} & {Some Values} \\
  \midrule
  2.3456 & 2.3456 & 2.3456 & 2.3456 \\
  {$\mathsf{a^2}$} & 34.2345 & 34.2345 & 34.2345 \\
  56.7835 & 56.7835 & 56.7835 & 56.7835 \\
  \bottomrule
  \end{tabular}
\end{center}

\end{document}

The second approach is to set the font used by siunitx for both text and math mode material to be sanserif

\documentclass[paper=a4, fontsize=11pt]{scrartcl}
\usepackage{booktabs,lmodern,siunitx}
\usepackage{siunitx}

\newcolumntype{k}{%
>{$\mathsf\bgroup}l%
<{\egroup$}%
}


\begin{document}
\sffamily

\begin{center}
  \sisetup{table-format = 2.4, math-rm=\mathsf, text-rm=\sffamily}
  \begin{tabular}{S
  k
  S[table-number-alignment = left]
  S[table-number-alignment = right]}
  \toprule
  {Some Values} & {Some Values} & {Some Values / \si{kg/m^2}} & {Some Values} \\
  \midrule
  2.3456 & 2.3456 & 2.3456 & 2.3456 \\
  {$\mathsf{a^2}$} & 34.2345 & 34.2345 & 34.2345 \\
  56.7835 & 56.7835 & 56.7835 & 56.7835 \\
  \bottomrule
  \end{tabular}
\end{center}

\end{document}

In both cases, I've used table-format = 2.4 as a shortcut to set up the reserved space. This is often easier than doing things piece by piece.