I had to change:
function start_lvl(&$output, $depth)
to:
function start_lvl( &$output, $depth = 0, $args = array() )
Because I was getting an incompatibility error:
Strict Standards: Declaration of My_Walker_Nav_Menu::start_lvl() should be compatible with Walker_Nav_Menu::start_lvl(&$output, $depth = 0, $args = Array)