Walkthrough
While making your way through the various Warp Tile Maze found in Silph Co, you will eventually make it to a section of the fifth floor of the building where Team Rocket Admin Archer can be found. Archer will be with another Team Rocket Member, the one that is actually in possession of the Card Key that can open all the doors within Silph Co. The two of them will challenge you and your Rival (who has added the Cubone from the Pokemon Tower to his team) to a Multi-Battle, and by defeating them, you will receive the Card Key.
Team Rocket Grunt
Level 34 Muk
Level 34 Raticate
Team Rocket Admin Archer
Level 37 Electrode: Self-Destruct, Thunderbolt, Swift
Level 37 Golbat: Air Slash, Crunch
Level 37 Weezing: Protect, Sludge Bomb, Flamethrower, Dark Pulse