What kind of sprinkler can I use on a lawn shaped like this?

The best irrigation system to use on lawns is a spray head system. The heads are designed to that you have head to head coverage and you have to use 4" pop up heads. It also depends on what type of lawn you are using because some varieties love to waste water while other do not. The beset thing to do is to have licensed contractor some to your property and test your existing system to make sure that it is working etc. Some time he may be able to retrofit your existing system without having to redo everything and sometimes he may have to redo it all. Good luck and hope this helps.