Interesting that you don't talk about ending the affair, which is the first thing you should do, and not just because you are being blackmailed; how would you feel if you partner was cheating on you? No matter how you met, she's dealing with her so-called 'difficult time' in a totally unprofessional and immoral way; how do you know things are stressful, she just told you? You're being used for sex, nothing more.
Second, you don't know that the blackmailer has video; they may have just seen you. Either way, contact the police because, as above, you'll lose however you chose to address it; demanding more money if you pay or starting the rumour anyway as they know you're desperate. The teacher is being quite brave involving the police as she could lose her job over it.