I also had the error message "The typescript compiler couldn't be found".
I had an older version of Web Essentials and when I installed version 3.7 the error message dissapeared but no js files where created.
Then in: Tools->Options->Text Editor->Typescript->Project
I checked "Compile on Save" (Automatically compile TypeScript files which are not part of a project) and then the js file got compiled.
Also when I look at the properties of the typescript file I now see TypeScriptCompile in Build Action. When I had the older version of Web Essentials I did not have the option "TypeScriptCompile" in Build Action.