Strongly recommended: Ferroseed, Litwick, Klink
Other options: Darmanitan, Pansear/Simisear, Vanillish if it knows Mirror Shot, Pignite/Emboar if you have it, Escavalier if you have it
Every pokémon that you will fight in this gym is a pure Ice type. As such, they are all weak to Fire, Fighting, Rock, and Steel moves.
How to get through the Gym Puzzle
This gym features ice floors, as well as curved walls that can be rotated by pressing a switch. The curved walls send you on a curved path across the ice, and you end up on a blue tile nearby.
Strategy for the Trainers
To start, go straight up from the entrance of the gym. When you end up on the blue tile, go to the right. This will put you near Black Belt Grant. He has two L35 Cubchoos and an L35 Cryogonal. For the Cubchoos, use Ferroseed and attack with a physical Steel or Rock move. For Cryogonal, try using Klink, Litwick, Escavalier, Vanillish if it knows Mirror Shot, or Pignite/Emboar if you have it.
After defeating Grant, go up and press the red button, then slide to the left to hit the curve and end up on a blue tile near Battle Girl Miriam. She has an L37 Cryogonal. As with the previous Cryogonal, use Klink, Litwick, Escavalier, Vanillish if it knows Mirror Shot, or Pignite/Emboar if you have it.
After Miriam, head up the stairs and jump over the ice ramp to reach Black Belt Kendrew. He has an L37 Cubchoo. This Cubchoo's moves are all Normal type, so bring Litwick into this battle and use its Fire moves.
After Kendrew, go to the right. When you bump into the rocks, go up. Then go to the right, bumping into the side of the curved thing. From there, go up. You will find Battle Girl Mikiko, who has an L36 Cubchoo and an L36 Vanillish. Use Litwick against the Cubchoo. Try using Klink or Litwick against Vanillish.
After Mikiko, go straight down from her to bump into the curved thing, then go to the right. Go around and slip off of the pointed tile. You'll end up on a blue tile. Go straight up from this tile to press the switch and rotate the curved thing. After that, go straight up, and then to the left to end up in front of Mikiko again. Slip off of the pointed tile, and the curved thing will put you on a blue thing to the right of some stairs. Go up those stairs to end up next to Battle Girl Chandra. She has two L36 Vanillish. As with the previous one, try using Klink or Litwick.
After Chandra, slip off of the tile to the left. Then go up from the pointed tile. After you land on the blue tile, go to the right, then up. Go to the right from the pointed tile. From this second blue tile, go up, then slip off of the tile at the top to end up near Black Belt Thomas. Use Klink or Litwick against his L36 Vanillish, and Litwick against his L36 Cubchoo.
After Thomas, step on the red tile to the right. Then go to the right and slip down off of the pointed tile. The curved thing will send you to a blue tile. Go left from the blue tile, then down. Then go left from the pointed tile to hit the curved thing. You will land on a second blue tile. Go straight up from it to reach the stairs to Brycen.
Strategy for Brycen
Brycen has an L37 Vanillish that knows Frost Breath (Ice Special), Acid Armor, Astonish (Ghost Physical), Mirror Shot (Steel Special). I recommend using Klink, or, if you have them, Cobalion or Escavalier.
Brycen also has an L37 Cryogonal that knows Aurora Beam (Ice Special), Frost Breath (Ice Special), Reflect, Rapid Spin (Normal Physical). Since it knows a Normal move, I recommend using Litwick against it.
Brycen's highest-level pokémon is an L39 Beartic that knows Swagger, Slash (Normal Physical), Icicle Crash (Ice Physical), Brine (Water Special). Because of its Water attack, Ferroseed might be a good choice since it is also strong against Normal attacks. You might also try Cobalion, Escavalier, or any pure Water type against it.
After you defeat Brycen, he gives you TM79 for Frost Breath, as well as the Freeze Badge. You can exit by going down the ramp to the left, which puts you near the entrance of the Gym.