Hmmm, well considering that both are immune to electric based attacks, this comes down to who has the better non-electrical abilities,skills and training.
Nora outclasses Cole in terms of training and experience, since she has been training to be a huntress for a few years, whereas Cole only had his powers for a few months. So she has the the edge there.
Nora also has the edge in raw strength, since she is on the GJ+ range and Cole is only in the GJ range. And she gets stronger when charged up with electricity.
They're about equal in terms of natural durability, since they are both MCB. However, Cole has two advantages. Cole's durability is Town level with shields, and he has a better healing factor, since it allows him to heal from bullet wounds in seconds and heal from a beating from David in just a few panels, twice.
Cole has the edge in range, since Nora's only notable long ranged weapon is her grenade launcher, which Cole can easily counter by using a shockwave to reflect the projectiles or maybe shoot them with bolts to prematurely detonate them. Cole has rockets, grenades and shockwaves that can travel quite a distance.
Also, all of Cole's bolts are now useless in this fight, since they are simply pure electricity and would do nothing but add onto Nora's strength. If Cole doesn't realize that all he's doing is empowering Nora quick, then it's all over for him.
Luckily, Cole doesn't rely solely on electricity to fight. His grenades, rockets and shockwaves are more focused on kinetic energy than electricity. It literally says in the games description of the moves that they "deliver a payload of kinetic energy," so they should be able to harm Nora, with only giving her a minimal power up, since they do use a bit of electricity. Combine this with his fire/ice abilities, and Cole now has a better shot.
Round 1: I say Cole mid-high difficulty. If he realizes that his bolts are useless early on, and uses his range advantage with his ice attacks and grenades, rockets and shockwaves, he takes this. Also, Cole can heal himself up and regain his stamina quickly, whereas Nora would slowly lose Aura as well as strength.
Round 2: I say Cole low difficulty. Basically for all the same reasons above, but now Cole has fire attacks and is WAY more ruthless. Combined with the fact he can turn intangible via Bat Swarm, he take takes this pretty easily. He can one-shot with Ionic Drain, and could possibly drain her blood, though I think her Aura could protect her from his bite.
Round 3: Cole high difficulty. Though he is forced to rely solely on his grenade/rocket/shockwave attacks, he can now regain energy much more quickly, and now has a barrier he that protects him from damage. He can also summon attack drones to distract Nora and give him time to breathe. And his phantasms can psyblast her, and they are immune to physical attacks, just touching them will blast your mind. Two of these almost killed end game Cole, so they are pretty deadly. His ground pound attack could throw Nora off her feet. So yeah.
