Translation:Even though I ask, she never forgives me.
Beg pardon, that is bad english because the first part of the sentence hasn't happened yet, requiring a future tense for the second part as in "she will never pardon/forgive me" (if it ever happens that I ask).
This sentence can describe repeated occurrences, all in simple present.
"Even if/when I ask, she never forgives me"
"Même que" is not a conjunction. You can only use it with the adjective "même" in a comparison: cette voiture est la même que la mienne (this car is the same as mine).
The conjunction is "même si".