Unfortunately, I don't think there is anything anyone can tell you that will make it better other than going to an ER/urgent care type setting (mind you this is coming from a health care professional).
It sounds to be a very infected ingrown toe nail. If after you soaked it in Epsom salts and tried to cut the nail out, you didn't succeed, the odds are a professional needs to take it out.
I am uncomfortable that your nail tech is trying to "correct" your problem. There are multiple varying factors here:
1) If your tech made an open wound and the foot bath wasn't clean to begin with
2) Most NSS (non standard salons) do not practice standard hygiene for sterilizing their tools. If a tool was used to cut skin that was not properly sanitized...
3) Your nail tech should probably not take it upon herself to correct your deeply ingrown toe nail, especially if you are not familiar with the process. It is a completely different story if the hangover nail is not down near the quick.
If you do end up going to an ER, which I suggest, you do need to take your bill to the salon, provided your toe was not in this state when you walked in for your pedicure.