The counterintuitive thing is that you can safely put your hand inside a tub of liquid nitrogen.
I still can't figure out how the liquid nitrogen managed to stay liquid in the cocktail and in her mouth and down her throat. It just doesn't make sense, unless it was encapsulated by something colder than -190 degrees... which can't be the case.
http://www.randombio.com/info/nitrogen.html
Swallowing liquid nitrogen
Yes, there are people who do tricks with liquid nitrogen. One low-temperature physicist makes a toast with liquid nitrogen. It is dangerous - don't do it.
The danger of toasting with liquid nitrogen is the possibility that you might swallow a bit by accident. If this were to happen, then the valve at the top of your stomach might close, trapping some liquid nitrogen inside you. Then some very bad things would happen:
The liquid nitrogen would be held in contact with your tissues for long enough to cause damage, possibly freezing your epiglottis or destroying the valve at the top of your stomach or some other part of your mouth, throat or stomach.
Second, the rapidly boiling liquid would generate a huge amount of vapor inside you. You might not explode, but it would be extremely unpleasant.
If she necked the cocktail even a relatively small amount of LN would be enough to cause damage.