It sounds like you have a wired headset and cut the cable that connects to your xbox instead of the cable that connects into the Xbox 360 controller. You can get both game and chat sound by reconnecting the wires back to their original connections, then cutting only the cable that connects into the controller. If the cable is a single "unit" then cut the cable below the volume adjustment puck on the controller side of the cable. If you are still having trouble I have a troubleshooting guide here:Ultimate Xbox One Headset DIY Adapter Troubleshooting and FAQ
The guide covers every issue people have come to me with so far. I'm also posting another video soon that shows how to connect the cable directly to the chat connector for a more professional/long term solution so you might want to wait for that video if you are handy with a soldering iron.
If you're still having trouble after that shoot me a message and I'll walk you through whatever part you are stuck on. Good luck! And sorry for any confusion in the tutorial!