She then appealed but was refused as 3 law lords said she had no grounds too, she had been treated fairly.. under the law, now you can argue that law is wrong but not that she was treated unfairly.
Some people think that because everyone appeals everything in US dramas (and I've read US attorneys grumbling about how that misinforms most of the US public) then everyone can shout "appeal !" even as the (never used) gavel is reverberating away.
The boring reality is that an appeal is only heard if there is point of law in question, or evidence can be presented to show that the sentence of the court may have been incorrect (either too harsh, or in some cases since they changed the law, too lenient).
You don't get to appeal the verdict. As the aforementioned Andrew Malkinson knows, courts are loath to overturn a jury verdict.
Just for the hard of thinking, the only person who convicted LC was LC herself. Once she entered a guilty plea (that we know she was fully informed about the implications and consequences thereof) then it was a relatively simple matter of the judge getting all the relevant facts in order to determine sentencing.
Appeals are not automatic. Not can they overturn the sentence.