My first Boston Terrier was attacked by a Belgian Malinois (similar to a German Shepherd) many years ago, so I can give you my opinion from the other side of the story...
While at the time I was so furious that this monster would DARE hurt my baby that I couldn't speak let alone think coherently, it was immeasurably better to have the owner of the Malinois come over and apologize. It was an accident, he was just as upset as I was, and eventually I did calm down and we ended up trying to make each other feel better.
Had he not approached me, I would have simmered and stewed and (possibly) eventually blown things all out of proportion.
In your case it was truly an accident, but since it was your dog that initiated the fight, it really does fall on you to take the first step and approach the woman. The note is a great idea but if she doesn't respond to it I would seriously consider going to see her. In your shoes I would want peace of mind that the issue was resolved to everyone's satisfaction rather than have it hanging over my head.
Talking to her (either in person or over the phone) will be difficult and awkward, but be patient. It's safe to say that she is also going to need to get up the nerve to call you and will probably be very defensive and critical. Expect her to say the worst at first, keep offering your support and understanding, and I'm sure you'll be able to work through this.
Good luck - you'll both come through this just fine.