Unfortunately, the origins of the name aren't known.
'Kathryn' is one of the many spellings of 'Catherine'. The name was first introduced into England by the returning Crusaders. It may have been popularised 800 years earlier (in the 4th century) by the martyrdom of St Katherine of Alexandria but, as I've stated, the original meaning name has been lost.
Aikaterine is a Greek given name for a female. Its origin and meaning are debated. It could derive from the earlier Greek name Hekaterine, which came from hekateros "each of the two". It could derive from the name of the goddess Hecate or the Greek aikia "torture". It could be from a Coptic name meaning "my consecration of your name".
The Romans falsely derived it from Greek katharos "pure" and changed their spelling from Katerina to Katharina to reflect this. Kathryn is a variant thereof used most often in Latvia, Ireland and England.