First, you are root-finding on the function
f[k_, w_] = -((w^2 Sqrt[25 + k^2 - w^2] (-2 + w^2))/((25 - 26 w^2 + w^4) Sqrt[(k^2 - 2 w^2 - k^2 w^2 + w^4)/(-1 + w^2)])) + Tan[1/2 Sqrt[-k^2 + w^2 (1 + 1/(1 - w^2))]];
Any time you are root-finding on a function that diverges to a zero in some denominator somewhere, numerically finding roots is going to be a problem. If there's some unlucky cancellation that occurs, there might be roots at points where the denominator is zero, but we can proceed as if this is not the case, and check our work at the end. Then, multiplying by the denominator (which is assumed to be non-zero) cannot change the roots of our equation.
To that end, let's define a new function that gets rid of the denominator:
f2[k_, w_] = f[k, w] Denominator@Together@f[k, w] // Expand // Simplify;
There are then two ways to go about finding the roots of this function. One way is to use FindRoot
, but my favorite is to use ContourPlot
:
ContourPlot[f2[k, w], {k, -2 π, 2 π}, {w, 0, 6}, Contours -> {0}, ContourShading -> False]

You can then extract the points from the graph using
pts = Cases[Normal@pC, Line[a_] :> a, Infinity];
and refine them using FindRoot
:
refinedPoints = Map[
Prepend[FindRoot[f2[#[[1]], w] == 0, {w, #[[2]]}, MaxIterations -> 10000], k -> #[[1]]] &,
pts, {2}] // Chop;
Then,
{k, w} /. refinedPoints // ListLinePlot

Finally, there's a little bit of trouble when we get to larger values of $\kappa$. To figure out what's going on there, we do the following:
PowerExpand@ComplexExpand@Normal@Series[f[k, w], {k, ∞, 1}]
Limit[%, k -> ∞]
Solve[% == 0, w]
N@%
which yields
(* I (-((2 w^2)/(25 - 26 w^2 + w^4)) + w^4/(25 - 26 w^2 + w^4) + Sinh[k]/(1 + Cosh[k]))
(I (25 - 28 w^2 + 2 w^4))/(25 - 26 w^2 + w^4)
{{w -> -Sqrt[1/2 (14 - Sqrt[146])]}, {w -> Sqrt[1/2 (14 - Sqrt[146])]},
{w -> -Sqrt[1/2 (14 + Sqrt[146])]}, {w -> Sqrt[1/2 (14 + Sqrt[146])]}}
{{w -> -0.979018}, {w -> 0.979018}, {w -> -3.6113}, {w -> 3.6113}} *)
so we can see the limiting values of $\omega$ at the wings.
ListPlot[Table[{κ, (ω /. Chop@FindRoot[eqn, {ω, 1.4}])}, {κ, -10, 10, 0.1}]]
$\endgroup$