The reason why I say you are lucky is your dentist is honest

He is honest.  There are many many ways to deny his responsibility on the infection.  Any germ or personal neglect in dental care can cause infection and it is very hard to conclude it is solely caused by the dentist.  This dentist is honest that he admitted his fault.   If he really denies you may end up with nothing.  Again, you don't have to agree with me but I have seen many similar cases.  That's why I tell you you are lucky.