Sounds like you just need to start over with both and remove all Homegroup(s). Then make a new Homegroup on one and join with the other.
Want to share the printer but don't want the to have to have the desktop turned on just to access the Internet.
I'm not sure what you mean by this statement. What does sharing the printer have to do with accessing the internet?
FYI: It doesn't matter which computer the Homegroup is created on (although a stationary desktop makes more sense because it is always there). As long as both are sharing printers, it should work.
Example: you start the Homegroup on the laptop and join the desktop to it. The printer hooked up to the desktop is shared with the laptop through the Homegroup and all is fine.
The opposite would work as well (start Homegroup on desktop, join with laptop, laptop uses printer that is shared by desktop).