Hello and welcome to JustAnswer! My name is XXXXX XXXXX thank you for giving me the opportunity to help you with your networking question.
Your TC-P42ST30 may have come with a wireless USB adapter - do you have this device? It would look similar to this:
Your adapter should work with the AirPort - let's find out why it isn't...
Make sure the adapter is connected to the TV.
Open the Menu on the TV, then select Network Settings. Below Connection Test, change the Network Connection to WiFi (Wireless). Select Wireless Network, just below Network Connection, then select No.
You will be at a screen that gives you an option to Search for a network, like this:
Select that option. The tv will search and should find your wireless network. Select it and enter your wireless password XXXXX prompted if you use one on your network.
Once this is complete, run the Connection Test. Any luck?
Strange - in the Network Settings, change IP Address/DNS to Manual, then enter these values:
IP address: 10.1.1.20
Subnet Mask: 255.255.255.0
Default Gateway: 10.0.1.1
Does the Connection Test work now?
What is the IP address you have listed?
No problem, glad to help.
On your computer, run these steps:
1. Open AirPort Admin Utility, located in Applications/Utilities.2. Select your base station from the list and click Configure.3. Click Show All Settings.4. Click Base Station Options, then select the "Enable Default Host at" checkbox.
Change the address from 10.0.1.253 to the one that matches your tv - 10.0.1.20. Save the settings, then unplug the Airport for 1 minute, then power it back on.
Power on the tv and run the Connection Test.
Let's back up just a bit...
Can you manually enter these numbers into the network settings of your TV and test please?
IP address: 10.1.1.10
If you find it, it is 10.0.1.1, isn't it?
What I was looking for on your computer was a screen like this to show the Router address: