unless rules have changed drastically, the bank generally doesn't apply for your kennitala. It's done by your employer.
Mine took about a month to get but I've heard stories of it taking up to six months. That was someone who was in a small town not accustomed to dealing with immigrants though! In Reykjavik it should be shorter.
As Mkay said though, finding a place to rent in a week is optimistic, but if you are reading through rental sites regularly and calling the instant a new listing appears, you might be lucky. Given that most rentals are on a small scale (i.e. no rental agencies), you should be able to work something out with the landlord if you don't have a kennitala on day 1. They already know most of the digits in it anyway.