On the other hand, why did we never say "Eastern Germany, Southern Vietnam, etc." or something similar for other split nations?
The name "Northern Ireland" was specified in the Government of Ireland Act 1920 which partitioned Ireland. I have never seen any documents suggesting "North Ireland" instead so I presume that there drafters saw it as grammatically correct. It was unusual for nationalists to use the official title, they preferred "the North of Ireland" or "the six counties". Sorry I can't give a more specific answer but it didn't seem to arise at the time.