Mathematica 12.0 can compute:
sol = Integrate[(JacobiDN[k*x, m]^2 - EllipticE[m]/EllipticK[m])^4, {x, -EllipticK[m]/k,
EllipticK[m]/k}, Assumptions -> {0 < m < 1, k > 0, EllipticK[m] > 0}]
(* -(1/(105 k EllipticK[m]^3))(630 EllipticE[m]^4 +
840 (-2 + m) EllipticE[m]^3 EllipticK[m] +
28 (61 + m (-61 + 16 m)) EllipticE[m]^2 EllipticK[m]^2 +
16 (-2 + m) (-5 + 2 m) (-5 + 3 m) EllipticE[m] EllipticK[m]^3 -
2 (-1 + m) (71 + m (-71 + 24 m)) EllipticK[m]^4) *)
Check:
f[k_?NumericQ, m_?NumericQ] := NIntegrate[(JacobiDN[k*x, m]^2 -
EllipticE[m]/EllipticK[m])^4, {x, -EllipticK[m]/k, EllipticK[m]/k}]
f[1, 1/2]
(*0.00551185 *)
sol /. k -> 1 /. m -> 1/2 // N
(*0.00551185 *)
Regards M.I