First.... you have done nothing wrong. You have to keep reminding yourself of that. It's hard not to blame ourselves when our child hurts another child.
This is my experience with biting.....
Usually a child bites when they are frustrated with something and aren't old enough to be able to use words to tell us. I think the best thing would be to start teaching her words to use instead of actions. If you notice it happening at a certain time of day... like when she is really tired before nap time, try giving her some extra attention at that point (before her melt down).... or let her watch a video alone to distract her.
I know it is frustrating.... I did in home daycare for years when my kids were that age. It is a phase though, and a normal one....but scary

I think the key is not to just give her the negative attention when she does it. If she is looking for your attention and not getting it... and then biting. She will see it as a way of getting the attention she wants(even if it is negative) and continue the biting. Does that make sense?
It is hard to tell a toddler you have to see to the needs of more than one child. They still think they are the center of the universe.
Good luck..... just remember this won't last forever