At first glance, evolving an Eevee into Sylveon might look a little confusing, but it's actually quite simple and shouldn't take too long at all.
Eevee → Sylveon
First things first, you're going to need an Eevee that knows a Fairy type move. You can either breed an Eevee yourself, or catch one in the yellow flowers on Route 10 that has the Fairy move Baby Doll Eyes.
Next, take out your Eevee in Pokémon Amie and raise it's Affection level to at least two hearts. You can raise Affection by feeding or petting your Eevee, or by making faces at it (a minigame best done in private). Once it's affection is at at least two hearts, battle with Eevee to level it up and it should evolve into Sylveon.