I've had two repair experiences with Nikon and both of them were good ones.
As you've found out, always keep the receipt. It is in effect your "warranty card". Nikon does not rely on mail-in product registration card for proof of warranty purposes because legally you aren't required to mail them in for warranty protection. The product registrations only do two things: 1) gives Nikon USA some marketing data on you, 2) gets you on Nikon's mailing list. In fact Nikon no longer includes the mail-in cards in many of their products. If you read the warranty info that came with your camera, you should see that it requires the bill-of-sale for service. I keep the originally packaging for my cameras and lenses and keep any unused items and the reciepts in them.
As mentioned, if you can't find it, try your retailer to see if you can get a copy. Lastly, you can try to call the service department at Nikon and try the old trick of "Screaming and begging". I know some people that have effectively used this technique with Nikon serivce staff. Appeal to their sense of logic. I had a friend whose D70 was out of warranty, but was exhibiting signs (the meter would randomly way over expose images) that I knew was the precursor to the well known "Green Blinking Light of Death" syndrome that Nikon was fixing for free as a "recall". Since technically his camera hadn't "GBLOD'ed" yet, they wanted to charge him for the repair. He called them and talked to a service manager who agreed that the attempt to charge him wasn't warranted and they performed the recall work for free.
As to the cost, Nikon uses tiered pricing for their repairs that may or may not bare a resembalance to the actual repair cost (like a lot of service departments at car dealers are doing). They usually classify a fix as a "A", "B", "C", etc. (i.e. "Minor", "Moderate", "Extensive", etc.) repair with each level having its own price.
and the repair should take 7-10 more working days. 
- Hope it's not that long...
