I wouldn't be attempting to control her by adding more pain to the equation. Ditch the chain bit & either put her in a double jointed snaffle(or something 'light'), or - my preference - get her reliable without a bit first. It is normal & understandable that she has reacted in this way to harsher treatment. She is reacting without thought *because* of it, rather than being able to relax & *respond*, which may well have been the basic problem to begin with. When they're reactive, they literally can't think clearly.
So first & foremost I would be ruling out any source of potential pain, as that is such a common cause of 'fast' horses. She is NOT "just being stubborn about it" for the hell of it - there is good reason behind her behaviour, whether you find it or not. Make absolutely sure she's comfortable in her saddle & see if she behaves differently bareback or in a treeless. Make sure her girth is not too tight. (Balance International have some good info on their site if you need more about saddle fit) Make absolutely sure her mouth is in good order, not hurting & the bit is right for her. Google Dr Cook bitless for some more info on not-so-obvious effects the bit may be having. Other potential pain issues include back generally - aside from saddle, sore, sick feet, neck issues & rider balance.
Make sure she has been taught well & is reliable about responding on the ground & to rein cues without a bit first, so you've at least got a foundation to begin with, rather than having nothing but force & pain to control her. Lack of trust/fear is a big issue for these prey animals, and again, adding pain/punishment to the equation doesn't help with that. Teaching her respectFULLY and clearly that she can trust & rely on you is important. Fear is another emotion that blocks rational thought & leads to an animal just reacting.
Perhaps all of the above is basically in good order, but she has been well 'programmed' by prior training/previous owners just to run everywhere - it is habit & she doesn't know how to do anything differently. Perhaps that's what your instructor sees as 'just stubborn'. I agree that lots of repetition & consistency is a big key, but it is still important to do it in a gentle, clear & respectFUL way. Therefore...
I would be starting off in a safe area, such as a small arena/paddock, so you can *teach* her persistently without having to force her for your safety. Eg. if you ask her to walk & she starts trotting, you can apply your gentle aids and just quietly persist, escalating gradually only to a level of *mild discomfort*, until she responds. She can't run off with you, so there is no need to be impatient & resort to force & pain which will cause her to react. BUT depending on her previous experiences & how ingrained her attitude, it might take quite a bit of patient persistence on your part to begin with, to convince her there is no need for defensive reactions. Be prepared to take as long as it takes.
Once you've established that she is reliably responsive to gentle pressure & cues on the ground, then I'd start riding her, pref. in a bitless/bosal/halter. Get her going reliably slowly first, before 'upping the ante' with speed; Teach her to stand patiently & calmly. Teach her to halt & turn from a walk with gentle seat, leg & rein cues. Teach her to back up & go sideways. Once she can do all that reliably & calmly from a standstill & walk, then teach her to do it at a trot. Don't attempt to canter until she's well under control at slower paces, but again, if she does break into a canter when you don't want, you're in a controlled environment, so can *ask* her & persist, rather than having to *force* her to slow.
Once you decide to take her out of the small enclosure, If possible I would stay in a safe environment such as a medium sized paddock, teach her to be reliable there before going out in the open. Remember, horses don't generalise well, so just because you've taught her reliably in one environment/situation doesn't mean she 'knows' what to do elsewhere. So be prepared to start at the beginning in a range of environments - tho once the basics are well established it will generally take far less time & effort elsewhere.