Printer not listed as a connected device in the router.

> I've installed a brand new N600 WiFi Dual Band Router.

   "N600" is a speed, not a model number.  Ever helpful, Netgear product
packaging emphasizes speeds like "Nxxx", but that's not the model
identifier.  Is the "WNDR3400v3" in the footer correct?     

> [...] a HP LaserJet Pro MFP pinter hardwired to the router,

   Also not an actual model number.  "hardwired" how?  Ethernet?  USB?

> I do not see the printer listed in attached devices.  It's plugged in
> and worked earlier today through an older router, so I know the cable is
> good.

   You know more about it than I do.  I don't even know if it's an
Ethernet or a USB cable.

   If it's an Ethernet cable, then the devices on at least one end of
the cable should have LED indicators which would be more reliable than
your opinion of how good the cable must be.  After that, actual
communication would depend on the printer's network configuration, about
which I know nothing.  If the printer has been given a static IP
address, then that might be compatible with the old router but not with
the new one.

   If it's a USB cable, then I would not expect the printer to appear in
the router's list of (network-)attached devices.  Especially if the
router is a WNDR3400v3, because, as its User Manual says:

      Note that the USB port on the router can be used to connect only
      USB storage devices like flash drives or hard drives. Do not
      connect computers, USB modems, printers, CD drives, or DVD drives
      to the router USB port.

   For more information, visit http://www.netgear.com/support , put in
your model number, and look for Documentation.  Get the User Manual.