Typing games really are fun to play. There are many of them which are probably too simple to play. There are dozens of typing games online. Some are free, others you can buy and download. Some good-quality typing tutors will also include great typing games, like Typing Master and other. Typer Shark indeed is a very famous speed typing game.
Some online typing services offer great typing games to play online as well. Usually these services will require paid registration for monthly or annual membership, but you can usually have a free trial to learn to type online with all their tools, including typing lessons, tests and typing games. Typing Web and Custom Typing are examples of this.
Personally, I'm not sure that a typing game will help you learn to touch type all the way, but a good one will likely help you significantly improve your typing speed while having fun.
My 2 Cents....http://www.online-typing-tutors-town.com/typing-games.html