(Ty puts his Law Student hat on)
Judge Judy is entertainment, it is nothing more than a weird type of arbitration with cameras, which I might add can be fairly easily appealed against so doesn't really solve anything. There is generally four steps when solving a disputes case like this, first there's negotiation, obviously you've pretty much gone through this. It is a good idea then to first try mediation. No lawyers have to be involved, (people can bring along a 'support person',) all you have to do is hire a mediator, (a mediator can be a retired judge, a lawyer - whatever you'll find plenty around.) What it is is pretty much negotiation but with a referee. At the end the mediator will summarise and give a ruling. This ruling is not binding but can be persuasive if the matter continues to the next level.
Arbitration as the next step is a more adversarial forum, most often lawyers are involved in this as is a judge. Arbitration is very much like litigation, (court,) but importantly it is cheaper and takes a lot less time. The judgement in this is binding but of course can be appealed if one party chooses to do so. Another important factor is that when one gets summons to appear at an arbitrary meeting unlike litigation one can choose not to. However I imagine both you and your sister want this solved so that shouldn't be too much of a problem. Arbitration is most similar to Judge Judy but without the argumentative bitch, (then again I've never met your sister so maybe you won't get off that easy. )
Your final option is of course litigation, the most adversarial forum. This of course requires a lawyer, court costs and most likely an incredibly long waiting period before your case comes up. Use this option only if there is no other. It's the most expensive but of course the decision is completely binding and the only way to appeal is to take the matter to a higher court - in a matter such as this it doesn't happen often where one party feels so strongly as to take it to a higher court. Litigation is also quite a public forum, but then again, not as public as "Judge Judy".
Now there is of course the chance she may sue you - problems like this should go through negotiation, mediation, arbitration and litigation but America generally likes to skip ahead right to suit. In New Zealand we don't sue at the drop of a hat, (I'm fairly sure at some point in America someone was probably sued over a dropped hat,) so it's not exactly my best area, but I'd say that if this is a real threat than Judge Judy would be the place where you'd be least likely to be forced to pay huge damages if you're found in the wrong - if you want to take the safe-yet-embarrassing route it's the best option. If you still don't want to take the Judge Judy bullet then hire a good lawyer and see if you can't get the case thrown out at first reading in good old normal court.
If she doesn't decide to sue and actually wants the matter solved as it should be my recommendation is to start with mediation. Hire a mediator, get yourself a support person, (a friend or spouse,) and see how that goes. From what I've read though this may not be the last step and you may have to move to arbitration, get a decent lawyer who has the ability to stay cool when faced by an angry bitch. I doubt that it will extend beyond this step.