It's highly unlikely that this is a result of any core files being changed. The menu item types is a dynamically created list based on the various views that are available and set up to be menu item types in the installed components.
The first thing you should do is go and remove any of the files that the extension you tried to install left in your file system. Most likely, one of those files is causing the function that builds the menu item types to break when it gets to that component. Remove the issue and your will likely get the rest of your menu types back.
Also, Lodder is absolutely correct. There is never any reason to edit a Joomla core file. You can accomplish anything you want through overrides and extensions, there is never a need to edit core files.