If you do something by mistake, it has happened accidentally. Not accidently. In Australia, we can partly blame the regular appearance of this error on 1990s pop combo Frente!
The band's 1993 single 'Accidently Kelly Street' reached #4 on the Australian charts, and sported a cover with the incorrect spelling prominently displayed. Reputedly this was a mistake by the graphic designers. but the band decided the error was charming and quirky enough to leave in place. The song has aged somewhat better than the decision to be wilfully illiterate.
It's true that when spoken, accidentally is often pronounced with the fourth syllable essentially omitted. That does not change the spelling when you write it down. Accuracy matters, and it's no accident.
Lifehacker's Mind Your Language column offers bossy advice on improving your writing.