If your Sylveon is Lvl 35 now, go to the move relearner at Mt. Lanikala(be sure you have a Heart Scale) and teach it the move.
Commented 35 minutes ago by Jason the Sly Nicvee
This pretty much sums it up. Since Sylveon only learns Draining Kiss at Level 20, this is the only way to get it to know Draining Kiss. Alternatively, I would just not worry about Draining Kiss and just wait until it learns Moonblast at Level 37. To get a Heart Scale, have a Pokémon with no held item that knows Thief, go to the Fishing Spot on Route 9, fish up a 70% chance of a Luvdisc, use Thief on it and pray for that 50% chance of it holding a Heart Scale comes true.
Hope that helps!