Oh, B.! Your post sounds so scared, you poor thing.
First off, no way do I agree that you failed her. My gosh, as soon as you asked her, she told you the truth. Even though you wish it weren't the truth, it is, and she trusted you enough not to lie.
Second, parents aren't in charge of their teenager's sexuality. She didn't betray you; she just took the lessons you'd taught her and made her own decision.
Believe me, I'd be a bit panicked in your shoes, too, so I'm not on any high horse about this! But I do know that when a 16-year-old decides to have sex with her boyfriend of a year, it's not anything to do with her mom or dad. It's because she's in love. And a huge part of being in love is sexual attraction.
Third, as soon as you found out, you decided to make sure your daughter is safe medically and take her to a doctor. (Planned Parenthood is a good option if health insurance is a problem, by the way.) You also admitted straight off that you can't put a stop to it if she chooses to have sex, so you didn't bury your head in the sand. You then asked all of us on this website for advice. My gosh, how in the world could you possibly think that you've failed your daughter?????? You're doing the best you know how to do!!
She just happens to be nearing adulthood and fell in love. Again, I'm not saying I'd be happy if it were my daughter. I'm just saying that it sounds to me like you're a darn good mother, and you need to not worry that her decision is somehow your fault. It just is what it is, same as it has been for billions of girls throughout history.
I am rooting for you and your family, B., and I feel pretty good about the idea that you'll come through this scary time just fine. Your daughter isn't 13 and unaware of how to protect herself from pregnancy. I mean, she still could get pregnant, because no birth control is 100% effective. But there's a huge difference between 13 and 16, and between a teenager who doesn't know about protection and a teenager who does know.
By the way, I was a pregnant teenager, and so were a few people I'm now close to. We faced challenges for sure, but our lives were NOT ruined. I chose not to have an abortion and not to give the baby up for adoption, but some of the people I'm close with made different choices. The common theme for all of us is that whichever decisions we made, our families were there for us. And that made us OK. Sure sounds like your daughter would have that, too.