Also think about your relationship with your manager. Would he or she have reason to think that you'll waste your last two weeks or distract your co_workers instead of doing what you can to make the transition seamless? If so, you may be asked to cut your two weeks short.
Why a Resignation Letter? In these digital times, why should we submit a resignation letter, anyway? A paper trail does seem old_fashioned, but in this case, it's a best practice. It makes your intentions clear, as well as the conditions. You're removing any ambiguity about the end of your time there.