The country has had mixed success - it failed to get a US-based Kazakhstan newspaper to remove stories based on the documents but did win a restraining order that the files must come off Facebook and other websites on which they were hosted.
In New Zealand, it sought a court order forcing Mega to reveal data that would be used to track down the hackers, apparently so it could sue them through the civil courts for damages.
The case in New Zealand, heard by Justice Simon Moore, was based on the Mega cloud storage website being one of those hosting the stolen files. He dismissed Mega's concerns over a breach of privacy or its objection it was being forced to assist an investigation by Kazakhstan.
Justice Moore said Mega would have to provide Kazakhstan with internet protocol addresses associated with access to the account holding the files, email addresses, contact information, account information and any payment information it might hold. He said the information was "neither particularly revealing nor particularly sensitive" and did not "carry the same weight of confidentiality as an individual's email or phone records".
"What the republic seeks is to triangulate the evidence sourced from the Mega [data] ... associated with the accounts which were used to upload the stolen documents on to Mega's website. It is expected that when this material is combined with the evidence the republic already has in its possession, the hackers, or those operating closely with them, will be identified."
Mega chief executive Stephen Hall said an appeal was being considered. "We're very concerned this repressive regime, which has an appalling record, is not just seeking damages but more repressive behaviour might occur."
He said there was no evidence the person with the Mega account - who had been contacted about the court action - had any association with any hackers.
Dotcom made about $30 million selling shares in the company he helped start in January 2013 and is no longer involved.
Last night, he said he expected the firm would provide the information.
The editor of the exiled Kazakh newspaper Respublika, Irina Petrusheva, said last year the documents had revealed "serious public scandals" concerning European politicians and lobbyists paid by the government. She testified in the US over concerns she held for the safety of those whose personal details were sought by Kazakhstan for use in civil lawsuits.
