"She doesn't speak English but she speaks Russian." vs.
"She doesn't speak English but she does speak Russian."

There's no difference in connotation between these sentences in English. The "but" is contrastive, so the "does" makes no difference. The only reason you'd choose either variant was if Russian had some sort of contextual relevance. For example, let's say my boss asks me to meet a visitor at the airport. I ask "does she speak English?", and my boss, who knows that I speak Russian, answers "She doesn't speak English but she [speaks/ does speak] Russian." There's not the slightest difference there.

The simple affirmative version would be "She doesn't speak English. She speaks Russian."