I would (and have) go the route of CGI with either HTML::Template or TT, using GD::Graph to produce the graphs. You may even want to try CGI::Application in conjunction with a templating system.

because what wasn't happening, and therefore wasn't in the log, was also important

This should be able to be done in SQL as well, making the db do all the work. Mysql provides the ifnull(colname, 0) command, which I'm sure there is an equivalent to in sqlite.
Your graph can then be produced with something like:

# Your SQL here.
while (my ($timestamp, $field1, $field2, $field3)=$sth->fetchrow_array
+() {
    push @timeline, $timestamp;
    push @data1, $field1;
    push @data2, $field2;
    push @data3, $field3;
}
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To assemble the data into arrays for graphing, and:

    my $graph = GD::Graph::mixed->new($x_size, $height);
    my @data=([@timeline],[@data1],[@data2],[@data3]);
    $graph->set(
              y_label           => 'Your graph title',
              title             => "$title",
            line_width      =>2,
            line_types      =>[1,1],
            skip_undef    =>1,
            box_axis    =>1,
            correct_width    =>1,
        fgclr    =>"black",
            legendclr    =>"black",
            x_label_skip    =>$x_skip
                 ) or die $graph->error;
    $graph->set_text_clr('black');
    $graph->set( 'y_number_format' => \&y_format );
    $graph->set_title_font(GD::gdSmallFont,12);
        $graph->set( types => ['area', 'lines','lines'] );
        $graph->set( dclrs => ['pink','blue','green'] );
        $graph->set_legend('your legend1', 'your legend2', 'your legen
+d 3');
        $graph->set(y_max_value=>($y_max));
        $graph->set(y_min_value=>(0.1));
    }
    my $gd = $graph->plot(\@data) or die $graph->error;
    binmode STDOUT;
    print $gd->jpeg(100);
}
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to graph it. Wrap it all in a CGI where you can input the necessary values (start time, end time, what kind of data you want, etc. ).