You need to use Ground pokémon here in order to have a chance of defeating Elesa. Ground types render her signature move ineffective, which will help immensely in the battle. Your main pokémon to bring to this gym are Palpitoad (the evolved form of Tympole) which is Water/Ground, a Sandile which is Ground/Dark, and a Drilbur which is pure Ground (you can find it in swirling dust in Wellspring Cave). To complete your party, bring Blitzle and Roggenrola.
If you're really patient, you can level up a Drilbur to level 31 to make it turn into an Excadrill, which has the best defenses against the Emolgas. However, it shouldn't be used against Blitzle or Zebstrika because of their Fire moves.
Getting through the Gym Puzzle
Nimbasa Gym features a large roller coaster system. To get to each trainer and eventually reach Elesa, you must ride on the roller coaster and press switches to change the roller coaster's track. To get into a roller coaster car, just press up when standing in front of it. To exit, press the down button while inside of the roller coaster car that is parked at an exit.
Strategy for the Trainers
When fighting the trainers in this Gym, you will encounter three Emolgas. One is L25 and the other two are L24. Emolga is weak to Ice and Rock moves. The ideal choice to use against Emolga, in terms of defense, is Sandile. However, Sandile's ground moves won't work, so it needs to have some Dark moves to be able to fight Emolga. Some other choices that don't have quite as good defenses are Palpitoad and Drilbur.
You will also encounter three Blitzles. One is L25 and the other two are L24. Being a pure Electric type, Blitzle is only weak to Ground moves. A Palpitoad or Sandile would work well here.
Elesa has two level 25 Emolgas and a level 27 Zebstrika. All three know the Electric type move Volt Switch, which has 20 PP. When a pokémon uses Volt Switch, it damages your active pokémon, then the pokémon who used Volt Switch will return to its owner and be switched out with one of the owner's other pokémon, unless that pokémon is the only one in the party.
If your active pokémon is not a Ground type, Elesa likes to use Volt Switch on every turn. Volt Switch could cut your pokémon's HP down to less than half of the maximum unless it is at a relatively high level. Because of this, it could only take two attacks to KO your pokémon, which wouldn't give your pokémon much time to damage the enemy, unless it is fast.
However, if you do use a pokémon that is a Ground type, then Volt Switch won't work at all. It won't affect your pokémon, AND, because of that, the opponent pokémon won't even be able to switch out. This is a huge help in the battle.
Elesa's pokémon all have strong non-Electric moves, though. The Emolgas have Aerial Ace, which is a Flying move, and Zebstrika has Flame Charge. These attacks probably won't hurt your Ground type pokémon as much as Volt Switch hurts non-Ground pokémon, however. For this reason, it is essential to use Ground type pokémon against Elesa.
The Emolgas' move sets are: Quick Attack, Volt Switch, Aerial Ace, and Pursuit. Zebstrika's moves are Quick Attack, Flame Charge, Spark, and Volt Switch.
I recommend using Sandile or Excadrill against the Emolgas, and Palpitoad against the Zebstrika. You can switch your Blitzle into the active slot if you need time to heal up Sandile or Palpitoad. Remember not to use Ground moves on Emolga because of its Flying type. You should use Ground moves on Zebstrika, though.
When you defeat Elesa, she will give you the Bolt Badge, which makes pokémon of level 50 and up obey you. She also gives you the TM for Volt Switch. She will tell you that she will get route 5 open for you.
For the quick way out of the gym, step on the switch to the left of Elesa, then ride the roller coaster on the left in order to get back to the entrance.