LOL - sorry but I had to kind of giggle, this is exactly like my sister and myself. I am 21 and she is 11 (I know, BIG age difference!) When I lived at home it was constant bickering and I know it drove my mom NUTS! I know from my point of view, my sister was just plain annoying me. I'm not sure what your girls argue about, and there is certainly a maturity difference between 11 and 13, but I am sure it's still similar is some ways. I know for me, my sister had some smart-mouth comment to say after everything I say and it drove me crazy, so I would say something back (not very mature, I know). You said your daughter thinks she rules everything, does that mean she is trying to tell your younger daughter what to do? What kind of things do they argue about?
Something that finally, for the most part, fixed things at the house was when I moved out, haha kidding (kind of), but before then I would just agree with everything my sister would say. She likes to be the instigator so if she would say something I would just say "yup", "uh-huh", "you know, you're right" and although it would make her mad (which I have to admit made me laugh a little), she would get bored with it and leave me alone. Of course, we will argue, she's hormonal and pushing her limits. Not to mention, we are sisters, it's going to happen. Most of the time now it is out of fun and we joke around. Try telling your daughter to just agree with everything her sister says (whichever one is trying to start a problem).
As for your youngest and yourself fighting and her being disrespectful to you, that is a whole other issue. What my mom did for me and does for my sister now, was when I was being mouthy tell me to go to my room until I could be nice, she would NOT talk to me unless I was being nice. Now, my room wasn't a fun place to be in, no TV, computer and I wasn't allowed to be on the phone when I was sent to my room for that. If her room is a place she would enjoy being sent to, send her to another place she won't be able to enjoy herself. I HATED being sent away and I would be mad at first but calm down because I realized that I would be sent back if I didn't talk nicely. I don't know if that will work for you, but it did for me and does for my sister. You have to be firm and NOT let her talk to you unless she is being repsectful. Eventually she will,hopefully, realize how she needs to talk to be able to get her point across. Also explain to her how it makes you feel that you two can't communicate on a good level and it always has to be an argument. My mom would cry to me because she was so sad that I was always being rude (I went through a bad phase), that really got to me, I hated to dissapoint my mom. But I also hate to say it, it's the age. Testing waters, finding yourself, etc. Try to make some alone time to spend with each of your daughters... movies, shopping, etc.
Sorry if I wasn't much help!! Hope things get better for you.