My lo's cord came off at 10 days. In the UK they tell you not to put anything on it at all, and to fold their nappies down below it to let it air dry. This way it usually only takes a week or so. My lo's stayed on longer because I had antibiotics during labour and this is a common effect. It was a little gunky in there when it first came off, but the health visitor said it was fine and to just keep it clean with cotton buds (q-tips). The gunky stuff became crusty and gradually came away like a scab and now he has a perfect bellybutton.
Oh and I was gross and put his cord in an envelope and popped it in his 'keepsakes' box - along with the positive pregnancy tests from 10 months ago. My Mum did the same thing with me, lol!