One way to look at this: When someone does not want to do something, their refusal can be seen as continuing into the present, when you are describing their refusal. The person did it (refused) in the past; you are explaining your understanding that they (still) do not want to do it.
I asked Sue to (do something), but she will not do it.
This is correct and understandable. Because the contraction for "will not" is "won't", that is one correct answer for your question.
The reason that "doesn't" is not correct: Her refusal is not actually ongoing, continually.
However, there are sentences in which present tense -- "does not" -- can be used:
I asked Sue to run with me in the Marathon on Sunday. But she doesn't run.
In other words, Sue is not the type of person who runs. It's a continuous condition, in the present -- it conveys a present, permanent condition.