Try this, after translating i\_L back into your subscripted variable name,
i\_L[t] = myLaplaceTransform[sys, t, i\_L];
Print[i\_L[t]/.v0_ + v1_*E^(v2_*t) + v3_*E^(v4_*t) + v5_*t -> v0 + v5*t + 2*Norm[v1]*E^(Re[v4]*t)*Cos[-Im[v4]*t + ArcTan[Im[v1]/Re[v1]]/Degree]];
I believe that prints exactly what you desired except for your little degree symbol. I fumbled around trying to get that little character in there and decided to leave that up to you.
If you look at that carefully you will see it is not actually checking that your two complex terms actually have the same constants. That would be bad if those constants were not the same. But it is possibly equally bad if the constants differed by even a single bit and thus the pattern match failed. I will leave the contemplation of numerical analysis of this to you.