amet, I am so sorry you are in this situation. It happens to more of us than people realise. :( It happened to me, and kept happening for over a year before I took the advice I am about to give you...
The first thing to say is that you are not a 'useless mum'. This abuse is not your fault - just as it would not be your fault if it were your husband/partner abusing you. It's not your fault - you are a victim here... But despite this, you are still the person who has to find the strength to stop it, which is unfair and hugely difficult.
You need to call the police if she ever attacks you again - tell her you will, and then do it. You need to think very seriously about whether to call the police this time. It sounds like this has gone on for a while, and if it was going to stop any other way, it would have done. Whether or not you want to call the police about what she has done to you, I think you should also tell your other daughter that you will back her up if she wants to make a police complaint. It is extremely serious that she has had her nose broken (of course you know that) and she deserves your back-up. She may be feeling very torn between protecting herself and protecting her sister. :(
It may help you to read some other past threads. I have picked some out for you that I hope will be useful...
This is the post where I first realised what was happening to me was domestic abuse.
More than a year later, I wrote a long post full of advice for someone else on this thread
This is a thread where a couple of other mums talked about similar problems with their daughters.
Here is a thread where parents talk about their experiences of calling the police when their DCs have been violent.
Last but not least, keep coming back here for support. Quite a few of us (too many) have been where you are now, and we'll be more than ready to support you, and be here to listen and not judge.