alkerfn,
That does help.
To get the printer(s) to work on your network at home means that you probably need to reconfigure the home 8610 printer to join your home network versus its current configuration for the office network.
You home network probably has a different name (SSID), assigned IP address range, submasking, etc. I.e., specific configurations used to keep network devices "networked" and communicating within their network.
Since the office is working nothing needs to be done there nor should you do anything there.
At home you will need to look at the router itself and get the make and model number. You also will need to have administrative rights to the router so you can look at (but not necessarily change) your home network configuration.
You can get some information from any working wireless computer on your home network. Are you familar with the Cmd (Command Prompt) and the ipconfig /all command?
What needs to be done (in general terms) is this: The printer must be enabled to use wireless, be told to join your home network by name, agree to have an IP address assigned by your home router, and be told what the router's address is.
Here is a link to the 8610's User Guide:
http://h10032.www1.hp.com/ctg/Manual/c04204791
Network settings are covered starting on Page 105. You may have to read a few other sections to understand the bigger picture.
However, you may already have enough information about your home network etc., to permit you to just use the printer menus to get the printer wirelessly configured for and to match your home network.
Page 109 explains manually configuring the printer for IP addresses. Step 6 on Page 110 is where your home wireless network data needs to be entered.