Keys are specific to the product. If I understand correctly, your laptop has Norton Anti Virus preinstalled in it. You will not be able to use the key specific to Norton Internet Security Product in the Norton Anti Virus product.
1. Wait till the trial period of the preinstalled NAV product (or) uninstall the preinstalled product
2. Download and install NIS 2010 from the below link
3. Use the NIS key you have and activate the product
If you want to uninstall the NAV and install NIS, then you have to use the removal tool and have the key handy for your NIS 2010 license. Here is the link for NRT
You can download the Norton Removal tool from http://www.symantec.com/symnrt
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Is the free 60 day Norton trial a trial of Norton Internet Security 2010? You can't use your key from NIS for NAV, nor can you use a key for NAV on a NIS product. They have to be both the same product and your license for NIS has to be one for more than one computer also.