****.
I have thankfully never been in this position, and if I were, I don't know how I would react... but to take a guess, I think I would do some incredible soul searching. I would think about if I loved this person through good times and bad, and now being the bad, could I do that. I would think if there would be any scenario in which I could not forget. This would be key if I couldn't ever forgive. To this, let's assume there are no kids involved, because kids would come first in any decision.
Once I had my thoughts figured out and played every role imagineable, I would confront BOTH of them, together, as a way to say, I caught you. If I have decided that I could not forgive her, I would pull her aside and try to get a response out of her. Of course she'll be embarrassed and emotional. So I'd give her a few days and then set a specific time and date to talk about it, for the date forces her and you to settle your thoughts. Then, if all is not well, she's gone.
If I thought I could forgive and forget, that my love was too strong, I'd pull her aside and ask her to think about a few things. First demand that the carpool end. If she couldn't do that, then you know her head isn't in the game. Then I'd ask her if she was committed. Could she be committed. And if she wanted. Answer. Now. You are either in or you are not. There is no inbetween. If it took her too long to decide, then she would be gone. It's time to get my life back in order and find that happiness back agian, because as a caring human, you and everybody else deserves to be happy. That there is no compromise. And if the other party wavers on seeing you through that, then you are better off without.
I firmly believe that you will be able to see through her eyes into her soul whether she can be truthful about her loyalty to you.
If the whole thing ends happy, then I'd take it upon me to figure the heck out what went wrong in our relationship that caused her to seek that kind of attention elsewhere.
As for the bike, I'd slap on a new cassette and leave the chain even though I know most places would tell you to change both -- and yes, I understand why. But after the first part, you might as well get both, because you might be putting a lot more miles on it than you imagine.
Just know that if you slap a new cog and chain on, it will be smoother for a while before the old bike rides like, well, the old bike. And if you don't like the ride or it doesn't treat you kindly, then you know it's time for a new bike that might ride a little smoother.
Just my opinion, of course. Sorry to hear.