Sentinels of the Multiverse is a cooperative card game that fits all of your needs:
I want to keep to the superhero theme to help this new player engage more with the story, since it is something they are in to.
The players in the game take on the role of superheroes who work together to fight a supervillain and their minions, all while dealing with an unstable environment that may help or hurt them. Each of the character's powers are reflected in the mechanics (e.g. Tachyon the speedster draws lots of cards, Ra the Egyptian god has a scepter and deals lots of fire-based damage).
Bonus points if the game has an electronic version that can enforce rules without us having to remember them all, at least until they feel comfortable with them.
There is an app version of the game for Windows, Mac, Android, and iOS that enforces all of the rules. It's not a rules helper, it's the whole game. After a a game or two, you'll definitely know the rules, but I personally think that the real game has a low enough learning curve that you should be fine without it.
In terms of complexity, I am probably looking for something more akin to Munchkin than Magic: The Gathering for a card game, or Forbidden Island rather than Arkham Horror for a board game.
Well if it's any indicator, both Munchkin and Forbidden Island are recommended for ages 10 and up and Sentinels of the Multiverse is recommended for ages 8 and up. Personally I think that the game is about as complicated as those two, but hey, the manufacturer thinks that it's simple enough to be played by younger players than those two games. While there are a lot of card effects, they tend to be pretty straightforward and since the game is fully cooperative, players can help each other out if they're having trouble keeping everything straight.
Your question title also mentioned the game being "simple yet deep." The rulebook rates heroes by complexity, so new players can play a hero with a lot of one shot effects while more experienced players can have depth by playing heroes with lots of ongoing effects that interact with each other. Also the fact that there are over a million combinations of heroes, environments, and supervillains (and more if you add the expansions) means that there is a lot of replay value and depth as you try different interactions between characters.