HP Printer Wireless Set up Issue

To restore network defaults

1. Press the Wireless button on the product.

2. Press the button next to the down arrow on the display until Restore Network

   Defaults is highlighted, and then press OK.

3. Confirm that you want to restore the network defaults.

Since the printer was associated with another router I would suggest to restore the defaults. Then proceed to make the connection to the T-Mobile WIFI. To make it "easy" HP had a printer software and process using a USB cable to transfer the WIFI info to the printer from a PC. Maybe you remember doing that long ago? 

According to the HP manual, you need the HP Photosmart Software CD and included USB setup cable to provide an fácil way to connect the HP All-in-One to your wireless network. The information starts on page 12. If you don't have the software CD you may be able to login to the HP support page and obtain that as a download. Since the printer is so old it might be they no longer offer it as a download. My guess is that printer software was Windows 7 when it released so unlikely they will have any downloads for software or drivers for it given the age.  If you installed the software on your computer before check and see if you still have it operational. If so then you can run through the steps outlined in the manual to get the settings into the printer. 

Newer printers seem to connect to the WIFI with more grace. The old ones that only support WEP or WPA authentication seem to be a bit more of an issue. If you want it to be more secure it might be time to upgrade or just set it up connected via USB to your computer and share it with other clients on the network for printing.