Thanks for the information.
If the AirPort Express is not configured correctly with the type of modem that you have, it will only allow one device to connect to the Internet, not two or more. Normally, up to 10 WiFi devices can connect the Express if it is set up correctly.
I suspect that it's not. We need to check the Router Mode settings on the AirPort Express, but you will need to have AirPort Utility installed on the iPad before you can do this.
I do not think that you will be able to install AirPort Utility for a PC on your laptop because it was designed to run on Windows 7. Apple never updated AirPort Utility for the PC after that time.
Unfortunately, kind of a Catch 22. You need to have the iPad connected to the Internet to download AirPort Utility, but the iPad cannot connect to the Internet.
Shut down the laptop and leave it off
Power off the modem completely by disconnecting the power cable from the back of the device or disconnecting the power adapter from the wall AC jack
Power off the AirPort Express by pulling it from the power socket
Power off the iPad
Wait a minute
Power up the modem only and let it run for at least a full minute
Power up the AirPort Express and let it run for a full minute
Power up the iPad, but leave the laptop powered off for now
Check to see if the iPad can connect to the Internet now
If it can, go to the App Store and search for AirPort Utility and download it. It's free
Report back on your progress