Simple. All Kenny has to do is get her hooked on something else. Namely, his penis. Or to be more precise: Fear of his penis. Basically, he just has to dress up as Mysterion and rape her. Constantly. Like, 24 hours a day, seven days a week.
Think about it. Ever see those lifetime movies where a woman gets raped and then can't leave the house? It's like those Tyler Perry flicks - all Kenny has to do is take Karen by force, and she'll be too busy praying to God to save her from her sin to get hooked on any sort of drugs.
I mean, think about it another way - if Batman raped you every day, you probably wouldn't turn to a stimulant like methamphetamine, would you? I mean, you're already jumping at shadows because you know he wants to park his batmobile in your batcave - you wouldn't dare make your life more complicated by adding an upper to the mix.
And if Karen tried to turn to a depressant, like heroin, then guess what? All Kenny would have to do is run a train on her in her narcotic fueled sleep. Guarantee she wouldn't end using again. Bam, normal life, the washing machine, the car, the compact disc and electric tin opener, good health, low cholesterol, dental insurance, mortgage, starter home, leisure wear, luggage, three piece suite, DIY, game shows, junk food, children, walks in the park, nine to five, good at golf, washing the car, choice of sweaters, family Christmas, indexed pension, tax exemption clearing gutters, getting by, looking ahead, the day you die. All due to rape.