I've been thinking about this...
The problem is that math "k" and math "kappa" are pretty much the same, except kappa lacks the upper part. And this is confusing.
You mention a "differnet font" on your question, but the problem as you found, is that "text fonts" don't mix with "math fonts".
So an alternative would be use a different math font.
Here we use the "Zapf Chancery" math font to get a different "k" (left side). But it would look ugly next to \ell, so following Emre suggestion, we can redefine ell to become chancery's "l"
\DeclareMathAlphabet{\mathpzc}{OT1}{pzc}{m}{it}
\renewcommand{\ell}{\mathpzc{l}}
\newcommand{\kell}{\mathpzc{k}}
would create the k, l on the left side of the image.
In particular image 1 was done with
\begin{equation}
2\mathpzc{kl}\ell = 2k\kappa l
\end{equation}
but defining/redefining kell/ell would make this easier to type. After all, the objective is to get a "math k" that looks different from kappa and this accomplishes.
If don't mind changing all math fonts, you can use
\usepackage[math,condensed]{anttor}
and get
for
\begin{equation}
2k \kappa \neq 2 l\ell
\end{equation}
Unfortunately I don't know what would the proper way to create a "letter command" (like \kell or \fancyk) to get only the k using antykwa font as above.
EDIT: I've found
\usepackage{fouriernc}
also gives the R-like k, but with a more modern look
\ell
to use the same typeface as the calligraphic 'k'?