Asking for recommendation letter to a Prof on New Year, deadline in a week
Should you ask? Sure -- there's nothing to lose, and it doesn't seem like you have a choice. How to do it politely? Be very apologetic, state that you're aware of how annoying it is to ask for this so close to the deadline (and presumably during holidays), describe any reasons for asking so late, and note that you will understand if your professor is unable to write. 

Personally, I'd refuse such a request unless (i) I knew the student well, (e.g. someone in my lab) or (ii) I had a letter already on-hand that I could easily tweak and submit. ("Easily" doesn't mean it isn't annoying, though -- all the web-based recommendation forms still have a lot of boxes to tick, etc.!)