http://ipinfo.io/json
Will return a json object that you can parse for the data.
Do this:
http://ipinfo.io/8.8.8.8/json
And you'll get the info for IP 8.8.8.8
Replace that with whatever IP you need if you want to use that.
They do have a limit of 1,000 per day without a subscription.
ipecho.net/plain will just return the external IP address for the client/user
It doesn't appear to have limits so I'd go with that if all you need is the external IP.
If you need the internal IP for the user I don't think there is a way to do this simply. You'll need to point the user to a website with instructions for Windows on how to do it. The typical way is to look at the results of the command ipconfig
There may be a way to set up a command line to do that or there may even be an exe utility to write that info to a file which you can obtain in your AppGameKit program, but to do that you'll need some sort of launcher for your game. A bat file might work.