What is the priority here? Keeping this under wraps from the wifey/family or the $10K
If it were me, I'd think the privacy issue is most important. If I could guarantee the extortion would be hush hush then I'd negotiate if it were a small sum, say $1000, however...
I am wary about blackmail and extortion - it seems to be a never ending cycle, and with some psycho lady like you're dealing with, who's to say she won't threaten you again in the future? You'll never have assurances that your payoff money will be the end of it. My stance would be tough and firm, even though it would stress the hell out of me, and say no payout = call the police and let them deal with it.
And for sure, deny deny deny. I agree that you may want to mention something to your wife about some crazy email you've been getting, and tip her off ONLY IF you feel the psycho will actually spill the beans. Wait until the last minute, but wifey should hear it from you first. No one likes surprises. (My wife found my
Ashley Madison account on her own, and she was pissed... took a lot of explaining to get out of that one)