Can any other wifi devices connect? Is the wifi light or 2.4G/5G lights on? There is usually a switch on the router to turn wifi on/off.
WPS needs wifi to be turned on.
When you connect the printer by cable, does the corresponding LAN light on the router turn on?
Does the router assign an ip address?
Can any device at all connect either by wifi or by cable?
Is this a new printer or has it been working before?
Other devices connect to wifi but the printer won't. Does the printer display any wifi networks at all?
The LAN light doesn't come on when connected by cable. Do you have another ethernet device to check that the port is ok?
Can you disable the wifi function on the printer when you try it with cable?
Is there any error indication on the printer? Either a light or something on the display panel?
If you take one of your connected wifi devices and forget the network, does your network show up on it?
Your router might be set to hide SSID names;the names won't show on wifi devices.
Do you have Archer 1600? Go to Basic>Wireless page and check Wireless Settings. Untick Hide SSID.