HP P1606dn printer wireless w/o connection to router

You basically have two choices:

1. Contact HP to see if there is a wireless adapter for this printer, or
2. Add an AirPort Express to your current network configuration. Configure the Express to join the existing AirPort wireless network. This will allow the Express base station's Ethernet port to be enabled so that you can connect it to the printer.

You have several options......each with advantages and disadvantages.

Option 1) Run an Ethernet cable from the AirPort router to the new location of the printer. You can run an Ethernet cable up to 300+ feet or about 100 meters with zero signal loss. Upside....always the most reliable connection. Downside....you have to run the cable, which may be difficult or impossible depending on your circumstances.

Option 2) Use a pair of Ethernet over Powerline (EOP) adapters to send an encoded Ethernet signal over the existing AC powerlines in your home. The theory with these devices is that you already have the wiring in place, you just need the adapters. Upside....usually simple to install. Downside.....not as reliable as a simple Ethernet cable due to possible signal interference on high voltage AC home wiring.

Option 3) Add an AirPort Express to wirelessly "extend" the network to the area near the printer. The Ethernet port on the Express is enabled when you do this, so you could use a short Ethernet cable to connect from the AirPort Express to the printer. Upside.....possibly an easy installation. Downside.....possibly a more difficult installation since the wireless signal will likely have to pass through a number of walls....wireless is not as reliable a connection method as Ethernet.

If you decide to try Option 2 or 3, it would probably be a good idea to understand the store's return policy in advance, before you buy. With EOP and wireless in general, you really will not know how well......or if....things will work before you try them out in your home.

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.....was able to print wirelessly when I could connect the printer to my Airport by cable

What kind of cable? Ethernet? USB?

What model of AirPort router do you have?

Are you still using OS X (10.5.8) as you indicate in your profile?