My daughter went through a phase where sorry was very automatic - particularly when it came to forgetting to do something she had clearly been instructed to do or had agreed to do when asked. The sorries became so automatic they were no longer meaningful at all. I'll admit that at the time she was five almost six or just barely six, but I spoke to her about the idea that sorry means, "I really didn't mean it and I feel bad that I did and I will do my best not to make the same mistake again."
I talked to her about how using a word too quickly, to easily or too often, actually makes the word less meaningful. I made it clear that "sorry" had an implicit promise not to do X thing again, and that when she said it and then made the same mistake again, she was now breaking a promise and hurting my sense of trust in her.
I suggest having a conversation like this - but not during the heat of a fight. If you find yourself going back and forth with your daughter, she isn't going to hear and retain any lesson you might offer as she is too intently wrapped up in her emotions to do so. Instead, I suggest each of you "going to your corners." Saying something like, "I am feeling very frustrated right now and need a little time to calm down and gather my thoughts" to your child is modeling good emotive communication (and is probably honest). Both of you can take a "time out" and come back to the conversation when cooler heads have prevailed.
If, after a discussion like that, she still doesn't apologize in a sincere way - next time it comes up, she'll have to suffer the consequences of lost trust. Sorry, when you really mean it, can actually be really hard for a lot of people to say (one of the reasons I am not a proponent of forced apologies or eye-contact). She'll have to answer the question, "how do I get my parent's trust back?" and she may find offering up a meaningful apology is the quickest and easiest way to do that, but she also might surprise you with an interesting alternative.
After I'd spoken about the promise part of an apology with my daughter, if she offered up a trite apology she clearly didn't mean or hadn't thought too much about, I said, "Should I believe you right now? What are you actually apologizing for and what promise are you making right now?" This, in essence, created a situation where she had to answer in the way Koert describes without actually forcing an apology. Once in awhile she just shrugged at my question (which to me, meant, she wasn't feeling sorry yet, was mad or whatever and needed some more time to mull things over more). Either way, I always made sure to say, "I love you" somewhere in the mix whether I was upset or not. She had to hear that question for awhile before she earned back my trust in her apologies (by showing me she meant them). Now that she has it, she does occasionally still has need to apologize for one thing or another, but a simple sorry suffices because her history shows me that she means it.
Since you end your question with: "If I say that didn't sound sorry, it will just start a long verbal fight between us and things only get worse and I never hear a nice voice." I'm going to assume fights are a problem and recommend How to talk so Kids will Listen, and Listen so They'll Talk It really does offer up some good examples and good advice as well as explain a little about why arguing just doesn't get anybody anywhere. Plus, it is funny. I'm sure the book has influenced how I've listened to and spoken with my students as well as my daughter, and it has helped the parents of many an adolescent I have known.
I am also a huge fan of The Seven Habits of Highly Effective Families and find the advice it offers up to the heads of household to be effective in helping parents plot their family's course and deal with the day to day stuff that happens in the trenches in a pretty simple way. Here is a blog about the basic concepts outlined in the book. We've also used The Seven Habits of Happy Kids with my daughter and will probably eventually use the ones for teens with her too.