I have not heard of the term "gaussian variance" before but I suspect you mean that you want to use the Delta Method (as it is known to statisticians) which is equivalent to the Propagation of Error (as it is known to some engineers and physicists). If k, l, and d are statistically independent of each other (or at least their covariances are zero), then you could use the following:

El[k_, l_, d_] := k l/(d^2 \[Pi]/4)
Eld[k_, l_, d_, kd_, ld_, dd_] := Module[{vk, vl, vd, f, \[CapitalSigma]},
  f = (D[El[vk, vl, vd], {{vk, vl, vd}}]) /. {vk -> k, vl -> l,  vd -> d};
  \[CapitalSigma] = DiagonalMatrix[{kd, ld, dd}];
  Sqrt[FullSimplify[f.\[CapitalSigma].f]]]

Eld[k, l, d, kd, ld, dd]
(* (4 Sqrt[(4 dd k^2 l^2+d^2 (kd l^2+k^2 ld))/d^6])/\[Pi] *)
Eld[1, 2, 3, 4, 5, 6]
(* (4 Sqrt[95/3])/(9 \[Pi]) *)