How to dissolve chalk (CaCO3)?
$\ce{CaCO3}$ is sedimenting when mixed with water, and therefore is not easy to remove with hot water. Nevertheless several methods exist that actually dissolve it that are better than vinegar:


You could add hydrochloric acid again to convert it into soluble Calcium chloride, according to equilibrium:
$$\ce{CaCO3 (s) + 2 HCl (aq) &lt;=&gt; CaCl2 (aq) + H2CO3(aq)}$$
You could use carbonic acid instead 
$$\ce{CaCO3 (s) + H2CO3 &lt;=&gt; Ca(HCO3)2}$$
this is the reason shells are getting dissolved by an increase of the amount of carbon dioxyde in solution:
$$\ce{CaCO3 (s) + CO2 (aq) + H2O &lt;=&gt; Ca(HCO3)2}$$ 
But I recommend Sulfamic acid ($\ce{H3NSO3}$), is frequently used for removing rust and limescale, replacing the more volatile and irritating hydrochloric acid, it may be a little more expensive but is certainly not toxic.