One option is to use the resource function CombinePlots. Will require some changes.

ktimepath1 =
  Plot[{Evaluate[k[t] /. solm]}, {t, 0, 200}, 
    PlotRange -> All, 
    AxesLabel -> {"t", "k"},
    PlotLegends -> {"k"},
    Frame -> True, 
    PlotStyle -> Thickness[0.01]] (* The two overlay exactly so increase thickness *)

ktimepath2 = 
  Plot[{Evaluate[k2[t] /. solm2]}, {t, 0, 200},
    PlotRange -> All, 
    AxesLabel -> {"t", "k2"},
    PlotLegends -> {"k2"}, Frame -> True, 
    FrameStyle -> Red,
    PlotStyle -> Red]

ResourceFunction["CombinePlots"][ktimepath1, ktimepath2, 
 "AxesSides" -> "TwoY",
 ImageSize -> 600]

Hi Nathan,

The plots being combined with Show each have a very small y range that is separated by a relatively large y value between the two plots. On a combined scale they appear as straight lines. You will have to add an offset or use a log scale, or... Really depends on what you are trying to convey by combining them.