Which of the following amines will produce an intensely...

The formation of an intensely foul-smelling substance when heated with chloroform and ethanolic potassium hydroxide is the carbylamine reaction, which is a characteristic test for primary amines (both aliphatic and aromatic).

The failure to form a stable diazonium salt at $273 - 278\text{ K}$ indicates that the amine is an aliphatic primary amine. Aromatic primary amines form stable diazonium salts at this temperature due to resonance stabilization, whereas aliphatic primary amines form highly unstable diazonium salts that immediately decompose to yield alcohols and nitrogen gas.

Evaluating the given options:
Benzylamine ($C_6H_5CH_2NH_2$) is a primary aliphatic amine. It gives a positive carbylamine test but fails to form a stable diazonium salt.
$p$-Toluidine and $p$-Chloroaniline are primary aromatic amines. They give a positive carbylamine test and form stable diazonium salts at $273 - 278\text{ K}$.
$N$-Methylaniline is a secondary amine and does not give the carbylamine test.

Therefore, benzylamine is the correct answer.

Answer: Benzylamine