Doughnut is the correct spelling and it dates to 1809:
http://www.etymonline.com/index.php?term...
You can blame 'short / slang' spellings of many words on advertising agencies and commercial enterprises, and this is how they 'sneak' into our lexicon.
Do you recall that American Airlines saved $70,000 by eliminating just ONE olive in salads in the 1980s. See #5 at:
http://blog.travelfli.com/post/149062765...
Although unknown for sure, how much money did Dunkin’ Donuts save by eliminating three letters from the more correct ‘doughnuts’ for the exterior signs on their outlets? Letters cost money. How much does each letter cost? How many Dunkin’ Donuts outlets are there? In short, sometimes correct spelling is sacrificed by commercial enterprises to save money.