How to calculate the amount of chlorine emitted yearly by a leak in the air conditioning system?
For the sake of correct formalism, here the answer as it should be given in the textbook:

The mass flow rate of the leaking substance, under the assumption of linear scaling, is
$$\dot{m}_{\ce{CHF2Cl}} = 12 ~\mathrm{kg\, month^{-1}} = 144~\mathrm{kg\, a^{-1}}\; .$$

The mass flow rate of chlorine, as contained in the leaking substance, is then calculated as follows:
$$\dot{m}_{\ce{Cl}} = \dot{m}_{\ce{CHF2Cl}} \cdot \frac{M_{\ce{Cl}}}{M_{\ce{CHF2Cl}}} = 144~\mathrm{kg\, a^{-1}} \cdot \frac{35.453~\mathrm{g\,mol^{-1}}}{86.47~\mathrm{g\,mol^{-1}}} = 59.04~\mathrm{kg\, a^{-1}}$$ 

For the end result to the question "how many kilograms of chlorine will be emitted into the atmosphere each year?", please note that the mass flow rate still has a dependency on time
$$ \dot{m} = \frac{\mathrm{d} m}{\mathrm{d} t}\; , $$
which means that we have to do an integration so we can calculate the mass:

$$ m(\tau) = \int_0^\tau \dot{m}\,\mathrm{d}t = \dot{m}\,\tau$$

For one year, we get
$$ m_{\ce{Cl}}(1~\mathrm{a}) = \dot{m}_{\ce{Cl}}\cdot1~\mathrm{a} = 59.04~\mathrm{kg}\; , $$
which means that your attempt at answering your own question were very good, on spot and totally valid.