Search widget doesn’t use searchform.php

I’ve just enabled my template to use widgets. Previously I manually added the code for the searchform to my template, but now I want to use the basic serach widget.

The strange thing is, the widget doesn’t use my searchform.php code. It seems to be using a basic form, with no class or id and a hardcoded <br> between input and submit.

Is there another place I need to change the contents of the search widget? Or am I doing something wrong?