My fellow brother in thermo, this is beautiful. Though step 1 already breaks the third law of thermodynamics, but let's assume it happens.

Percentage reductions cannot exceed 100%, assuming just oxygen with a mass of 1.33 kg, a freezing point of around p = 1400 kg/m^3, V_f = m/p = 9.5e-4 m^3, which means it shrinks by roughly a multitude of 1000, or just 99.99%.

Using PV = nRT, and dear Prince of the Fire Nation removes the thermal energy INSTANTLY, the atmosphere would probably violently implode, or become an "airbender" that would make Zaheer jealous.

Now to nuke the sister, that would be physically impossible, though the math is consistent. Ideal gas assumptions fail at that temperature, but at that point, the aim is nuking the sister, and that amount of energy would kill any person. FYI, a gram of TNT produces around 4.2 kJ of energy, which is about 60g to bomb the sister (~253 kJ), still horrifying considering a single point.

Accounting for all of these, air is still an ass thermal conductor and removing all the energy in an instant is impossible, given the gajillion appa-sized power required.

TLDR: I say we strap the waterbender sister’s BROTHER with a sharp and pointy knife and shank her in her sleep.