The net ionic equation is:
[Ag(NH₃)₂]⁺ + Cl⁻ + 2H⁺ → AgCl + 2 NH₄⁺
As i can see that you balanced it correctly also you calculated the charge of each part correctly but you need to know more details about complexes:
complex is formed between Metal cations as Ag⁺, Cu⁺², Fe⁺³ and ligands which may be neutral as NH₃ or H₂O or negatively charged as CN⁻ or halide
The number of ligands attached to metal is the co-ordination number, in our case Ag has coordination number = 2 so it must attached to two ligands which are NH₃ here (neutral ligands)
- The charge of complex is the sum of metal and ligand charges +1 for Ag and 0 for NH₃ so the overall charge of complex is +1
so the complex must be attracted to negatively charged anion which is Cl⁻ here and both react with H⁺ from acid (HNO₃) to give AgCl precipitate and NH₄⁺ ion.