Role of NaI in Formation of Silyl Enol Ethers using Chloro(trimethyl)silane
The use of $\ce{NaI}$ in $\ce{MeCN}$ (a solvent in which $\ce{NaI}$ is quite soluble) is forming $\ce{Me3SiI}$ from the $\ce{Me3SiCl}$. The reason for doing this is $\ce{Me3SiI}$ is significantly more reactive with the triethylammonium enolate and will increase the yield of the TMS-enolate.

You are correct in thinking that the white fume you see when adding the $\ce{Me3SiCl}$ is $\ce{HCl}$. The product of aqueous hydrolysis of $\ce{Me3SiCl}$ is $\ce{HCl}$ plus hexamethyldisiloxane (Wikipedia):

$$\ce{2Me3SiCl + 2H2O -&gt; 2HCl + Me3SiOSiMe3}$$