I wouldn't use liquid nitrogen. I would use liquid carbon dioxide, CO2. There are a few reasons for this.
CO2 can exist as a liquid at ambient temperatures under sufficient pressure. Nitrogen cannot as its critical temperature is much lower. In layman's terms, at any temperature above 126K (-147C) the density (and other properties) of nitrogen gas and nitrogen liquid become identical, so there is no distinction between gas and liquid and no vaporization cooling occurs when pressure is released. So, if you use nitrogen, your dragon is going to need a way of storing liquid nitrogen, and a means of generating it.
CO2 on the other hand, can exist as a liquid at ambient temperature if sufficient pressure is applied to it (56 atm at 20C). When the pressure is released through a simple valve, molecules break free from the attractive forces in the liquid, which requires energy and therefore causes cooling. This is exactly what happens when a CO2 fire extinguisher is used. (though the idea is to suffocate the fire, a side effect is the production of dry ice.)
Fun with fire extinguishers (don't try this at home!) https://www.youtube.com/watch?v=Z3xyqfCZmSU
When CO2 vaporizes it requires 3 times as much energy per unit mass than liquid nitrogen, therefore its cooling effect at ambient temperature would be greater. Ironically, it is precisely because of this that it has a higher boiling point temperature than liquid nitrogen: -78C at atmospheric pressure instead of -195C. (As an added complication, the freezing point of CO2 is higher than -78C, so it can only exist as a liquid when its boiling point is raised by high pressure.)
Chemical / biochemical
Where does the dragon get her gas from? If it is nitrogen, she will have to get it from the atmosphere and liquefy it in some way, either via an expander or via a separate refrigeration cycle, both of which seem biologically impossible.
If she uses CO2 she can generate it chemically, and it may then already be under pressure. She may do this in a variety of ways:
- Use the normal metabolism. It is unlikely that it would be possible to build up high pressures of CO2 in this way, though, without making the blood too acidic.
- Use CO2 from her fire breath, if any. This would be highly inefficient, and would again have the problem of generating CO2 at ambient
- Go to a frozen wasteland and eat CO2
Eat chalk, and use this to generate CO2, through the reaction CaCO3 + Acid = Ca salt + CO2. The acid could come from normal metabolism.
As for 4, except the acid comes from an external source. For example, the dragon may go to a volcano and feast on brimstone (sulphur) which she could then burn (producing fiery breath without the inconvenience of having to generate large amounts of fuel through metabolism.) When she is not breathing fire, the dragon may slowly convert the sulphur to SO3 and H2SO4 in her belly, and react these with chalk to make CO2 under pressure. This is an entirely realistic way for a dragon to produce icy breath without metabolism or mechanical engineering issues. The only remaining issue is materials, which have always been a problem with fire-breathing dragons anyway.
Other suitable gases include propane (again requiring more energy per unit mass than nitrogen for vaporization than nitrogen, and I can say from experience that a splash of propane "feels" colder than nitrogen.) The dragon could choose to breathe icy propane, or instead ignite it and breathe fire.
Carbon monoxide, CO is another posibility (highly toxic and moderately flammable as well as being a gas) but its critical point is well below ambient temperature, like nitrogen. Several existing organisms use formic acid as a weapon, and formic acid decomposes catalytically to carbon monoxide and water.
Sulphur dioxide fits in with the brimstone idea. It has a boiling point of -10C, which is perhaps a little high, and is toxic and corrosive. Generating SO2 chemically under pressure would be difficult because of the requirement for atmospheric oxygen, unless the dragon breathed in the air and dived to great depths like a sperm whale in order to compress the oxygen.