Now that yall are talking about it, the best time for The North to have declared independence would have been during Robert's Rebellion. The same exact guys (The Greatjon, Karstark) that were around for Robb Stark and egging him on to make The North its own kingdom were around back then......it seems out of character for them to support a non-Targaryen King
The argument The Greatjon used was that Torren Stark only bent the knee because the Targaryens had dragons and that they would never follow no fakkits like the Lannisters.
So why would they follow a Baratheon
Lord Rickard Stark, Brandon Stark and Lyanna Stark had all died/been killed. Seems like a perfect time for The North to band together and declare independence
Robert was in no position to make demands. Robert needed The North to win his war. Ned put in just as much work as Robert did to win the war. Ned should have been like: "I'll help you get the Iron Throne but, The North is mine"