How do i display the built-in gallery inside a widget?

Widgets don’t usually allow shortcodes in them (they treat them as normal text).

What you have to do, is to tell this widget, it should run do_shortcode on it’s text. To do it, you can use this snippet:

add_filter('widget_text', 'do_shortcode');

Just place it in your function.php file. Then you can place shortcodes in Text Widget and they will work normally.