Unless otherwise stated, a spell lasts for it's duration.
The only rule that covers general spell duration is a small section on page 203 of the PHB.
A spell's duration is the length of time the spell persists. A duration can be expressed in rounds, minutes, hours, or even years. Some spells specify that their effects last until the spells are dispelled or destroyed.
And the rules that cover monster death aren't very extensive either.
A monster usually dies or is destroyed when it drops to 0 hit points. For more on hit points, see the Player's Handbook. (Monster Manual pg.7)
Most DMs have a monster die the instant it drops to 0 hit points, rather than having it fall unconscious and make death saving throws. Mighty villains and special nonplayer characters are common exceptions; the DM might have them fall unconscious and follow the same rules as player characters. (PHB pg.198)
Other than that, there's no specific rule that says spell effects end on a creature when it dies, unless the creature is the one casting and the spell requires concentration (PHB pg.203), so they should carry on (exceptions might be made at DM's discretion).
So a direct answer to your question would be: the creature would stay Enlarged for the full 1 minute duration of the spell, whether dead or alive, unless your concentration is broken.