I agree with Silas in that it's exceedingly unlikely that the Khanid would have maintained and utilized their original language after being subsumed by the Amarr. Sure, they weren't conquered, but even worse for their cultural heritage they were seemingly eager to join the Amarr and would probably have adopted the culture of their new masters ASAP.
That said... my main theory for why the Minmatar were able to rebel is that the Amarr didn't actually put any effort into exterminating their culture. Because the Udorian culture merged with the Amarr, the Khanid and Ni-Kunni more or less joined the Empire willingly, and the Ealur were cave-dwellers, there was never any real need to launch cultural purges of the societies that Amarr absorbed. In fact, it could be argued that the Empire's history has more counts of cosmopolitan tolerance than inquisitorial hounding. However, one of the big recurring themes of the Empire is its patience. All the Empire has to do to eradicate a culture is to conquer it and then simply stop recording its existence. The media and major archives of Amarr are controlled by those who perpetuate the core culture. A Khanid or Ni-Kunni who wanted to research the history of his race simply wouldn't find any cultural documents preserved. At most he might find census documents, historical land boundaries and genetic data. Any great treasures of culture would have been annihilated by time itself, including coherent languages.
Personally I'd argue that any Khanid language that did exist has been absorbed into Amarrian/Amarish. One could argue that there may be efforts in the Kingdom to linguarchaeologically restore the original language, but it's important to remember that the cultural elite in the Kingdom are still True Amarr. While expressions of bloodline khanid pride may be tolerated or even encouraged to blossom to build popular morale, I don't think it would be in the interests of the royals to let it get too far. Also, King Khanid II is a progressive man, not a regressor. He'd be more likely to reform the language to be less stuffy, more modern, and more useful.
But, if we separate the Kingdom from the debate, we know that the bloodline khanid are known to be cyberknights and religious zealots. These are extremes of dedication, devotion, passion and interest. It suggests an overarching trait within the khanid race for extreme focus on the pursuit of a single field or topic. Thus it's not unreasonable to assume that there could be bloodline khanid cultural revival groups out there, even though there's no actual faction to outright support them. And because of their intense focus, it's not impossible that these revivalists may have reconstructed a modern version of the ancient khanid language.
Anyway, if you haven't looked at it already, check out the Amarrad project in my signature. It's my project to build a stuffy "Old Amarrian" language that predates the Moral Reforms, and includes fragments and traces of the absorbed Udorian, Khanid and even Takmahl languages.