How strong a transformer do I need to convert from 220V to 11V at 1.6A?
You are right, a 30W step-down transformer should be fine. I suspect that the person at the electronics store was multiplying the input voltage by the output current, which is not the right thing to do.

Edit: Watt's Law says that power is equal to the product of voltage and current. (We will ignore the issues of reactive power and power factor for this simple case.)
$$ P = V \times I $$

On the output side of your charger the nameplate says that the maximum output voltage is 10.8V and the maximum output current is 1.5A. So, the maximum power out of the charger is 16.2W (watts).

On the input side you are given voltage and power, so we can calculate the current:
$$I_{IN} = \frac{P_{IN}}{V_{IN}} = \frac{26}{120} = 217\,\textrm{mA}$$

Thermodynamics tells us that the power out can never be greater than the power in, and in this case we see that this is true.