nowplaying.include.php Will Not Display Results

Your script is at /wp-content/themes/responsive-child-theme/nowplaying-example.php but you are including /nowplaying-example.php. That is not going to work. You need to provide the complete path for that include:

include(get_stylesheet_directory().'/nowplaying-example.php');

Assuming I have read that right, and assuming that the files are actually there.

Your other code– the widget code– contains curly quotes. Don’t use those. Use ordinary, boring straight up and down quotes like in the first code you posted. That could be causing errors. But that other code also has a bad include. Those bad includes will trigger warnings, which could cause other issues.