Take half the marinade and the steaks and place in a ziploc bag. Massage and let marinate for the day. (Cover the remaining marinade and set in the fridge until cooking.)
Warm a wok over medium-high heat and add 1 Tbs. of oil. Remove the meat from the marinade, letting any excess liquid drain off.
Add the steak to the wok and stir-fry just until it begins to brown, 2 to 3 minutes.
Remove the steak and set aside.
Warm the remaining tablespoon of oil in the wok, then add the mushrooms, broccoli, garlic, ginger paste, and onions.
Stir-fry until the vegetables begin to soften, 2 to 3 minutes.
Add the green onions and kimchi and stir-fry and additional 2 to 3 minutes.
Add the reserved sauce and the beef, stirring well to coat.