Maybe something like this will help.
neq20 = (BesselI[0, k/\[Lambda] r] BesselK[1,
k/\[Lambda] \[Rho]] +
BesselK[0, k/\[Lambda] r] BesselI[1,
k/\[Lambda] \[Rho]])/(BesselI[1, k/\[Lambda]] BesselK[1,
k/\[Lambda] \[Rho]] -
BesselI[1, k/\[Lambda] \[Rho]] BesselK[1,
k/\[Lambda]]) /. {\[Lambda] -> 800, \[Rho] -> 10}
(* Out[151]= (
BesselI[1, k/80] BesselK[0, (k r)/800] +
BesselI[0, (k r)/800] BesselK[1, k/80])/(-BesselI[1, k/80] BesselK[1,
k/800] + BesselI[1, k/800] BesselK[1, k/80]) *)
TrigToExp[Normal[Series[neq20, {k, Infinity, 1}]]]
(* Out[153]= -((
40 I Sqrt[10] E^(
k (-(1/80) - r/800)))/(((40 Sqrt[10] E^(-9 k/800))/k - (
40 Sqrt[10] E^(9 k/800))/k) k Sqrt[r])) + (
40 Sqrt[10] E^(
k (1/80 - r/800)))/(((40 Sqrt[10] E^(-9 k/800))/k - (
40 Sqrt[10] E^(9 k/800))/k) k Sqrt[r]) + (
20 Sqrt[10]
E^(-k/80) (E^((I \[Pi])/4 - (k r)/800) +
E^(-((I \[Pi])/4) + (k r)/800)) (r + Sqrt[r^2]))/(((
40 Sqrt[10] E^(-9 k/800))/k - (40 Sqrt[10] E^(9 k/800))/
k) k Sqrt[-I r] r) + (
20 Sqrt[10]
E^(-k/80) (E^(-((I \[Pi])/4) - (k r)/800) +
E^((I \[Pi])/4 + (k r)/800)) (1 - Sqrt[r^2]/r))/(((
40 Sqrt[10] E^(-9 k/800))/k - (40 Sqrt[10] E^(9 k/800))/k) k Sqrt[
I r])*)