Hello again Sam and thanks for writing back,
My name isXXXXX'm a licensed Texas attorney. I am honestly sorry for your circumstances, very much appreciate your patronage, and am glad to try and help out.
Accordingly, I'm pleased to say that the law is indeed on your side here. Here's how this works. It's a matter of common law (cases) as opposed to statutes. Contract law provides that each party has certain duties or responsibilities and certain rewards or benefits. Here, your duty is to pay the amount owed. You have fulfilled your duty. Your reward, so to speak, is for your sister to receive the diploma she is due, having fulfilled her end of the bargain by completing her coursework. The private school has already received its reward by virtue of the monies paid. Its duty, which it is not fulfilling, is to give your sister the diploma that she has earned. That's called a material breach of contract, it's entirely actionable (meaning filing a lawsuit) under a legal theory or doctrine called detrimental reliance and unjust enrichment. In fact, what you're outlined constitutes a rather classic case. I would suggest you mention all of this to the school (feel free to print out this information). If it still persists in not doing the right thing and withholds the diploma, you have ample grounds to have legal counsel make a formal demand of your behalf.
If you have a follow-up question or need clarification, please just say the word by using "reply" to reach me.
I truly hope all works out for you and your family.