The fastest solution would be waxing as suggested. If you can't get nads or it's too inconvenient to use there are many other products on the market. I would approach talking to her in a very sensitive way as she either might not have noticed or may already be worried about the hair. If she is like me she is probably stuck with waxing for life. My hair refuses to die. A good thing probably but I sure would love to give up shaving.
Another solution is one of the many home electrolysis products on the market. I only know of these because I have considered them for my bikini line but I'm a chicken when it comes to pain so I've not tried them. I would research dangers of the use of the products on the internet before making a purchase.
The healthiest solution is to find another type of birth control. They are very bad for women and I hate to think what they might be doing to her body if she is growing hair because of them. You are on your own on that one as that is such a personal decision. Do some research and discuss with her all the options. Weigh the damage this might be causing her body with what a unplanned baby might do. Hard decision but I'm sure if you are here you are caring enough to look for solutions.