I don't. I see intercessory prayer as a way of satisfying one's urge to do something without actually doing anything.
As the old line goes, "all that is necessary for evil to succeed is for good people to do nothing." Well, prayer takes good people's honest intention to do something good and turns it into literally nothing.
When I'm in an unpleasant situation, I don't want my feeling of powerlessness taken away unless it's because I can actually do something to help things. Just soothing my feeling does nothing but kill my motivation to effect real change for the the better.
If your friend was right there in front of you and it would hurt her to see you not praying, then that might change things, but I personally never feel the urge to pray for prayer's sake.
However, I do have some sympathy for you as a former Catholic surrounded by Catholics. Sometimes, old habits die hard, especially if you're still immersed in them in the society around you. And if we demanded that every atheist act like a "perfect" atheist (if such a thing even exists) all the time, then nobody would ever leave religion for atheism.
Plus, it's not like anyone behaves perfectly rationally when they're under major stress. I wouldn't worry about it too much.
I attend religious weddings, baptisms etc, and stand at the appropriate times, but that's about it. Even that much acknowledgement of ritual is enough to have my religious relatives accuse me of insincerity, and one atheist friend accuses me of being a faith enabler, but I just see it as respect for my family's right to believe what they like, rather than a flaw in my own position.
Your prayer, particularly the fact that you couldn't just say you'd done it, is about your friendship. You did something you knew had no worth and felt silly doing it, simply because your friend asked you to. Don't let anyone pull you up about it - theists seeking to find gaps in your case for your atheism or atheists picking at the boundaries you set for your atheism. Your prayer made you uncomfortable, suggesting you might refuse similar requests in the future, but your behaviour is your responsibility and yours alone.
I'm not going to tell you what to do. If you feel an obligation to pray, that is your decision.
However, ask yourself this: Which god are you praying to? Why? I presume you are praying to the Christian God. Even if prayer did work, why are you presuming it is the Christian God that is receiving the prayers? Is there any more reason to believe in the Christian God than, say, Allah, Thor, or the Flying Spaghetti Monster? If you converted to Hinduism and became ill, do you think your friend would pray to Ganesh if you asked her to do so?
Back when I was shedding my spirituality I always tried to be respectful about prayer. I went from bowing my head with eyes closed and reciting the same words aloud, to bowing my head with eyes closed and reciting the same words in my head. Next I just bowed my head and closed my eyes. Then I was just bowing my head. A little further down the road I stopped doing that and just kept my head up with eyes open. And now when it is time for the family to pray before a meal I actively take a step backwards and just wait for them to be done.
It's so uncomfortable. For the longest while I felt stupid with my head down and eyes closed. Even just standing there being part of the group I felt wrong. I realized that by trying to respectfully fit in I was ultimately betraying myself.
I think it is important to be yourself and be true to yourself. While I won't participate in prayer, I don't stop others from doing their thing. I just remain quiet and think about something else, or my husband and I stare in each others eyes. When it comes to the topic of prayer, I try to not make a big deal of my lack of participation when in a group and I don't acknowledge prayer requests in person or in text.
Prayer is therapeutic. It may not be answered by a supernatural entity but sometimes when people are confused or emotionally distressed, prayer helps to clarify one's emotions and give clarity. Journaling also helps.