I plug my Sony Clie PEG-NX60/u into the cradle, and after letting it sit for a while, I attempt to turn it on and a green power light shows. However, nothing appears on the screen, even if I leave it for a few minutes. Could it be a problem with the screen itself? And if so, where can I get a replacement screen?

Since the battery is within the unit it can leak or go BELOW the threshold level of the cell wherein the charger will not work to charge the unit. However if you can confirm the battery voltage to be below, use a multimeter to check, then before replacing use a JUMP START with an external limiter DC bench supply , you can use a series resisitor of about 47 ohms with a 6-9 VDC.
If the battery takes up voltage the see if the charger will respond and the set will power.

Plug it into the cradle, make sure the green light on the cradle is on.
With it still in the cradle, hold down the power button, then press the reset button on the back while still holding down power button.
Initialization screen comes up, memory is cleared, do a sync to restore memory.

You will have to restore your data afterwords.

It requires a wireless lan card that is sold separate.

Are you referring to the T615C?
You probably need to replace the internal battery.
Try a hard reboot while it's plugged into the charger first.

I found there is an online tutorial which said to hold down the power button for a few seconds and the light comes back on.

There's a switch on the side that says 'hold'. Move it to the lower position.

The unit will turn power on for the WiFi when you attempt to use Mail or the NetFront browser.  When the WiFi is active there will be a little green light in the bottom left corner of the unit.
For WiFi to work, you will have to setup the security key to connect to secure WiFi networks.  These settings are entered into the NetFront browser Network preferences.

Either the stick is defective or the wrong format.

I had that happen last week. Just let it dry out and when the display started jumping, I did a HARD reset. Synced and it's now working like a champ. (Mine had a lot of moisture in it but did not have total emersion in water.) Good luck.

I do not have a solution for you but if you cannot get it working I can use the case to repair my NX60, I think. I hope that you can get your unit fixed but if you cannot please let me know if you want to get rid of it. See my posting: Sony CLIE PEG-NX60 Handheld Sony PDA repair NX60/U I have a Sony PEG NX60.....