Love, I utterly agree with you telling this woman to STFU as far as your DD is concerned. You can try nicely (but firmly) and get blunt if you have to. it's THAT important. This woman has no business saying ANYTHING. Your DP needs to tell her that too tbh, but of course it's going to be hard.
If YOU state your boundaries, explain that until she stops insulting your DD, stops saying things that are inappropriate in her company that you will not be opening the door to her.
Once YOU have said that, all HE has to do is say, pumpkin has said she's unhappy with the treatment of DD, and I support her. All he need do is hold the family line (YOUR collective family)
If the woman refuses to stop insulting your DD, then she will stay away from the home. regardless of you telling your DD about her father.
HOWEVER.... your DD needs to be told about the facts of life (otherwise it will be a shock and she will need the time between now and then to process.) Your family may be 'late-starters' but there will be other girls in her class that aren't. SO she'll need to understand the subject, and WELL, or she may well be teased, (1) for not starting with the others and (2), being clueless as to what's what.
Honesty is VITAL, both in paternity and in life stuff. You are now about to make serious parenting mistakes if you continue to shove everyone's head into the sand. You may be a parent, but one that refuses to arm her with really big information is WRONG. She DOES need to know, and YES, NOW, and you will harm her development if you don't.
She will suddenly realise that she is NOT actually full sister to her siblings, that her dad is not her dad, her GP likewise.
She has a lot to deal with anyway on either subject, if you allow hormones to creep in too before you come clean, you will open up a MASSIVE, potentially off-the-rails kind of response.
TBH, as interferring as your DP's mother is, her making you post this thread WILL serve you well. BUT YOU HAVE TO TALK TO YOUR DD.