Problem with liquid helium is the latent heat of vapourisation is bugger all. The heat capacity of the vapour is huge, which is where all the cooling power is. Liquid nitrogen has a huge latent heat, and bugger all heat capacity of the vapour
That is why you precool a superconducting magnet with (cheap) liquid nitrogen, and only go from 77K to 4.2K with helium.
The final conclusion is: cool the drink with liquid nitrogen directly. Using it to cool balls of some sort, and then use them to cool the booze just adds a further step, and loses cooling efficiency. Just sling the liquid nitrogen straight in - a sort of nitrogen sling