[Executive summary: solve the underlying causes of your problem by becoming Pope]
I think it's a mistake to focus too much on the case of one particular convert to Catholicism simply because you know her personally. To do that is to fall prey to the availability heuristic.
The root cause of your problem with your friend is that the Catholic Church exists as a powerful and influential organisation which continues to promote its weird dogma, polluting Leah's mind along with the minds of millions of others. Before investing time and effort trying to flip her back to the side of reason, you should evaluate the costs and benefits of destroying the Church as an effective entity. I will now outline a method by which you and around 20 like-minded friends could do just that.
The Catholic Church is based in a tiny pseudo-state called Vatican City State. It has no permanent population and no true army, the Swiss Guard being more of a ceremonial bodyguard force (although they do have modern firearms as well as the cool-looking pikes).
What I propose is that you wait until the current Pope dies (not long now!) and a conclave has been assembled, then rush Vatican City in an infantry-style terrorist assault. There are 150 or so of the Swiss Guard but you could divide their forces by having some of you occupy a building, display simulated explosives and make fake demands. Your true targets are the cardinals who are there to elect a new Pope.
Once you capture the cardinals, simply force them at gunpoint to elect you Pope. In the event that you're not already a Bishop and therefore not an eligible candidate for the Papacy, simply mount a privilege escalation attack, whereby you force them to elect you to successively higher offices until you become a valid Pope. I anticipate that this process will be completed before the Italian state can mount an effective special forces operation to kill you.
Now you are Pope, you are the sovereign of the Vatican City State. You can pardon your co-conspirators, and appoint them as ambassadors so they have diplomatic immunity outside VSS. You can then use your papal infallibility to remove all the problematic doctrines of the Church (homophobia, opposition to birth control/abortion, etc.) and bring all the child rapists it has shielded and enabled to justice. Or you could change all Catholic doctrines to those of Pastafarianism. Either way, the appeal of Catholicism to your friend would be destroyed as its so-called timeless moral insights are revealed as human constructs. One or (preferably) more "True" Catholic churches will arise to challenge your claim to the Papacy, causing decades of damaging, hilarious schisms, during which you should make sure to declare several Antipopes.
I suggest you treat this post as if it's a joke, and then seek military training as soon as possible.
In the event that you're not already a Bishop and therefore not an eligible candidate for the Papacy
In principle, they're allowed to elect any baptized male (you'll be ordained bishop right after the election if you're not already one) , though it's been centuries since the last time the new pope wasn't already a cardinal. (Don't ask me what happens if they elect a married man.)
I recently learned that a friend of mine, and a long-time atheist (and atheist blogger), is planning to convert to Catholicism. It seems the impetus for her conversion was increasing frustration that she had no good naturalistic account for objective morality in the form of virtue ethics; that upon reflection, she decided she felt like morality "loved" her; that this feeling implied God; and that she had sufficient "if God, then Catholicism" priors to point toward Catholicism, even though she's bisexual (!) and purports to still feel uncertain about the Church's views on sexuality. (Side note: all of this information is material she's blogged about herself, so it's not as if I'm sharing personal details she would prefer to be kept private.)
First, I want to state the rationality lesson I learned from this episode: atheists who spend a great deal of their time analyzing and even critiquing the views of a particular religion are at-risk atheists. Eliezer's spoken about this sort of issue before ("Someone who spends all day thinking about whether the Trinity does or does not exist, rather than Allah or Thor or the Flying Spaghetti Monster, is more than halfway to Christianity."), but I guess it took a personal experience to really drive the point home. When I first read my friend's post, I had a major "I notice that I am confused" moment, because it just seemed so implausible that someone who understood actual atheist arguments (as opposed to dead little sister Hollywood Atheism) could convert to religion, and Catholicism of all things. I seriously considered (and investigated) the possibility that her post was some kind of prank or experiment or otherwise not sincere, or that her account had been hijacked by a very good impersonator (both of these seem quite unlikely at this point).
But then I remembered how I had been frustrated in the past by her tolerance for what seemed like rank religious bigotry and how often I thought she was taking seriously theological positions that seemed about as likely as the 9/11 attacks being genuinely inspired and ordained by Allah. I remembered how I thought she had a confused conception of meta-ethics and that she often seemed skeptical of reductionism, which in retrospect should have been a major red flag for purported atheists. So yeah, spending all your time arguing about Catholic doctrine really is a warning sign, no matter how strongly you seem to champion the "atheist" side of the debate. Seriously.
But second, and more immediately, I wonder if anybody has advice on how to handle this, or if they've had similar experiences with their friends. I do care about this person, and I was devastated to hear this news, so if there's something I can do to help her, I want to. Of course, I would prefer most that she stop worrying about religion entirely and just grok the math that makes religious hypotheses so unlikely as to not be worth your time. But in the short term I'd settle for her not becoming a Catholic, and not immersing herself further in Dark Side Epistemology or surrounding herself with people trying to convince her that she needs to "repent" of her sexuality.
I think I have a pretty good understanding of the theoretical concepts at stake here, but I'm not sure where to start or what style of argument is likely to have the best effect at this point. My tentative plan is to express my concern, try to get more information about what she's thinking, and get a dialogue going (I expect she'll be open to this), but I wanted to see if you all had more specific suggestions, especially if you've been through similar experiences yourself. Thanks!