Get printer to be wireless without a Wi-Fi button
Solution 1:
So do I understand correctly - you're connecting some random USB-Ethernet adapter to the printer, and wire from there to the router? Does the printer natively support Ethernet connection over USB port?
As is I doubt this has any chance of succeeding. The reason is standards and protocols - Ethernet works on 802.3, USB doesn't. It's like you tried to instruct a swahili speaker over phone how to bake bread... if you don't have a common protocol (language) it's not going to happen.
When you connect this kind of device to a computer, you have to install a driver (or Windows might do it for you). But I don't know any way of installing a USB network card driver to a printer.
What you need is a USB (or Wireless) Print Server. That's a small, fairly inexpensive piece of hardware designed for exactly this - sharing USB printers over the network. You connect USB to printer, wire to the router, and after a while you will see an available network printer.
Solution 2:
According to the manual for your specified printer, it is not a wireless printer. This leaves only one option remaining, as your router does not have USB utility. Wireless print servers are effectively replacements for the computer you don't want to have connected and leave on.
It's a device connected to power (obviously) and to your printer via the standard USB connection. The device is configured to communicate with your network (read and follow all label directions) which will then allow the printer to appear when you use the control panel "search for printers" option.
Amazon has many options with prices a few dollars either side of US$40 that will likely work for you.