The easiest way I found is to just use the "https://kahoot.it/" as the embed URL, then you create the kahoot on a different tab, to get people into the room you have to tell everyone the 6 digit code, which is mildly annoying.

I figured out that in the game options you want to enable "Show Questions and Answers on Players Devices" that way you don't have to describe everything that is going on and the colors that mean each answer.

Hi! It seems that Kahoot does allow embedding, but you need to follow their instructions on how to do so closely. You may need to make that a feature request you send to Kahoot asking them to make the process easier!