By RAW, yes. So strictly speaking, absolutely.
You've actually answered your own question citing the pertinent rules.
Requirement met: An enemy of the target within 5 feet.
That's really what matters. The target treats the illusion as real by default, so it's clearly an enemy of the target, which grants you sneak attack. Since this portion is based on the enemy's point of view, and not your own, you most definitely qualify.
Furthermore, illusion spells generally default as real until investigated/interacted with. Example:
PHB. pg.276, Silent Image
You create the image of an object, a creature, or some
other visible phenomenon that is no larger than a 15-foot
cube. The image appears at a spot within range and
lasts for the duration. The image is purely visual; it isn't
accompanied by sound, smell, or other sensory effects.
This does not in any way limit the creature from appearing as normal. It just lacks sound, smell and touch. Against virtually all creatures, an utterly silent drow warrior suddenly appearing behind them would most definitely be constituted as a threat.
Contrary to other answers, I would definitely not give the creature a free check against the spell. If there was meant to be a saving throw against the spell, it would have included it in the spell description.
The Arcane Trickster is already limited in spell slots and progression, and is burning rounds and spell slots to cast these abilities. The check most definitely needs to be a decision made by the creature to investigate the threat. Imposing a manufactured save would be detrimental to the player, and possibly be viewed as meta gaming against a rogue type for playing to it's strengths. This would be like making a barbarian pass an Acrobatics check before making a Strength/Athletics roll to open a door.
One answer contrasted this against phantasmal force in an effort to emphasize the reality of the spell. The actual difference between the spells, is that phantasmal force can actually harm a target (up to and including instant perceived death on a failed save). Whereas silent image can not harm a target directly, and any physical interaction reveals the image to be an illusion automatically.
As a personal note: I just want to caution you with respect to illusion spells. By default, illusions appear real unless otherwise stated or disbelieved by the affected target(s). This can be done any number of ways that is normally detailed as conditions within the spell itself that would end it. Not a lot of DM's handle these well because they know an illusion is being cast. The affected creature doesn't though, and for all intents and purposes, this is a spell which is resolving in an effect. Whether that be summoning a creature, a sword, a pool of lava or anything else, the illusion is real until otherwise countered from the affected creatures point of view.