I'm going to get my co-worker fired because she was pretending she was dating me to her family
What did she tell her family?
She was lying to her family about being in a relationship with me.
If no harm has happened is it really important you tell on her? I don't think it is worth the embarrassment and hurt it might cause by telling other people. You should talk to her.
I just want to teach her about honesty.
It is good to want other people to be honest. But there might be a cost if you report it and start more drama.