Hi, Sam.
First of all, your LiveID is the thing you need to reinstall and activate. You don't need to product key. To confirm that you know the correct LiveID before doing any reinstalls, log in at http://billing.microsoft.com and view your subscription status for OneCare. As long as you see the subscription, you've got the correct LiveID.
If you don't wish to have either PC as a hub, you'll still see the "Your OneCare Circle" link on the main OneCare screen. All PCs activated with the same LiveID/Subscription ID will be part of the Circle automatically, The Hub allows you to see the status of the other PCs in your Circle and allows you to set up and use Centralized Backup.
If the Printer is attached to the desktop PC, you should be able to use the Printer Sharing feature of OneCare, but you can also share it normally through Windows.
However, since OneCare appears to be broken on the laptop, I suspect that it is preventing this from working properly.
On the laptop, try running this tool:
Firewall repair tool - http://forums.microsoft.com/WindowsOneCare/ShowPost.aspx?PostID=2468376&SiteID=2
If that doesn't resolve the Firewall problem, please contact support.
How to reach support (FAQ) - http://forums.microsoft.com/WindowsOneCare/ShowPost.aspx?PostID=2421771&SiteID=2
-steve