Mr Jones was also pleased with how the appeal went but said he was surprised and angry to discover the police had failed to disclose a document which the defence lawyers said was a "revelation" to the case.
According to the document - which was not disclosed to defence lawyers at Lundy's original trial - the neuropathologist, Dr Heng Teoh, said Lundy should not be convicted on the forensic evidence he viewed.
"It's taken 12 and half years to find out something that should have been disclosed back in 2001. And it wasn't. So we are a little angry about that - it could have changed the whole outlook of how all the evidence was presented."
The couple are flying home and plan to visit Lundy at Rangipo Prison on Sunday where there would be "big hugs," said Mrs Jones.
Lundy's lawyer David Hislop QC, a New Zealander living in London, said he was "hopeful" the Privy Council would find their arguments more attractive than the Crown.
"There is a certain exhilaration at putting your best foot forward.
"I think this is a very strange case. When you've been doing these cases for as long as I have, you don't get yourself bogged down with guilt or innocence.
"But one develops a feeling that something may have gone wrong in this case. When you think that someone is sitting in prison year after year and might even die there. It's a pretty chilling thought."
He said he had a feeling that "something wasn't right" when he read the case file two years ago.
"We'll see whether their Lordships [the Privy Council] agree or not."
The appeal before the five members of the Privy Council, including Chief Justice Dame Sian Elias, is likely to the last of its kind for New Zealand.
The decision on whether to quash Lundy's convictions will not be released for several weeks, if not months.
The Privy Council could deny the appeal outright but there were two other options.
The Law Lords could order a retrial or send the case back to the Court of Appeal, which could hear the conflicting evidence over the disputed brain tissue evidence and then decide whether a new trial should be held.
The appeal hinged on a number of key planks submitted by Mr Hislop which included:
- the failure of police to disclose a document to defence lawyers.
-the validity of science used to identify matter on Lundy's shirt as brain tissue.
- the examination of stomach contents to determine the time of death.
- the examination of a computer's shutdown time which led to allegations Lundy tampered with the machine to create an alibi.
- the failure of the judge to properly direct the jury about the evidence of the sole eye witness.
Deputy Solicitor-General Cameron Mander responded to the criticism on behalf of the Crown and also reminded the Privy Council of important evidence in the original trial.
Stains and smears of Christine Lundy's blood on a window was a strong suggestion the window was forced open after the murders occurred as a "staged scenario" of a burglary, said Mr Mander.
The sliding door beside the window was left open.
Fragments of paint flakes found on or near the bodies matched samples of paint on Lundy's set of tools, said Mr Mander.
In closing his submissions, Mr Mander said if the Privy Council was to grant Lundy's appeal there needed to be another hearing in the Court of Appeal to determine the strength of the scientific evidence offered by the defence lawyers.
In response, Mr Hislop disagreed.
"We respectfully submit that if the criteria of fresh evidence is met, if credibility is met, then the dispute between scientific experts is a matter for the true finders of fact - the jury."
