The trick here is to realize that ammonium hydroxide,
If you write the complete ionic equation for this reaction, you'll get
The net ionic equation for this reaction is
The ammonia reacts with the silver cations to form the diamminesilver (I) metal complex. To balance this equation, simply multiply the ammonia molecules present on the reactants' side by 2.
To get the initial equation balanced, simply multiply
Finally, balance the hydrogen and oxygen atoms by multiplying the water molecule present on the products' side by 2.
And that's how the balanced chemical equation looks like.