Ok cool i get it.
Heres what I do with chicks like this. The anal ones (somtimes turns out ironically HAHA) you have to take it slow!
She is a stiff. You need to relax her and make it less proffesional. But obviously you cant from the begining. You almost had it but i reckon your neg was a little too harsh. And very obvious that you were conveying interest (which wouldnt normally matter but since its email its totally different). So we know why you fucked up there. First of all you have to get her off her high horse.
1. When you meet, shake her hand, also use your left to clasp over it. Just that little more to make her think 'is he hitting on me?' the basic theme of this is to make her think shes 'seeing things' SO THAT SHE CAN BLAME THEM ON HER GOING CRAZY! Also (later on) you need to make sure she knows you can take all the blame.
2. Eye contact, sometimes this is hard, my old english teacher hb 7.5 who was honestly one of my best teachers i for some reason could not hold e.contact with her! She has these big blue eyes and a smile it was so hard not to look away. You gota do what my 5ft 5 little blond english teacher did to me. You have establish dominance over her. Let her know, she isnt in complete control.
So you want her to know that she isnt in control BUT its STILL IN CONTROL. If you get what i mean?
E.contact is very inner game and alot to do with believeing that it isnt awakward, because for somepeople its not. Just writing about my english teacher brings VIVID memorys of the feelings i had etc. EC is very powerful and memorable. For the next few minutes if you did it right she will be thinking about it. Glance up when your working (perhaps in a time when your working and shes just sitting there working on her own stuff) and wait for her to make EC when she does smile like you won. And carry on working still smiling.
Once you break that barrier of professionalism by doing these things you can start with a opener. Its important tho that you give her an excuse to talk to you in this way. Cus she wants to but doesnt know if she should..and there all naughty girls lol. So more situational convorsation on things like maths and what got her into tutoring in the first place to warm her up. What intrests you in maths add in things when you say this that DHV, your driven and a motivated worker etc...
As soon as all this starts its important (because of limited time) that you build rapport quickly with her. Pace and lead its called, and you start by mirroring. Rapport is hugely associated with body language. Look at the way shes sitting, is she leaning in or out. Does she have one arm on the table or none? Mirror her but dont mimic her you dont wana freak her out. Usually this happens naturally (ie when you walk with your friends and you all walk in unison). Then mimic her breathing rate. She should naturaly open up to you. Now that you have her in rapport youve finished pacing now you can lead. So once you know shes there you can slowly move into a more comfortable convo engaing body language position
Things should go smoothly from there.