Any module that counts visitors per day?

You can always use Google Analytics for that, there's a handy GA module available.

I had the same problem and some similar. First I shout "Hurray" when I discovered the Visitors module and then I began to curse about its limitations. (Well, I had also this Google Analytics friends, which I kicked in the ass). I end up copying the whole module in the subdirectory "mymodules", renaming it to "visitors_extra" and writing my own filters which I linked in an extended menu. My code now looks as followed (only the relevant snippets):

....
 $items['visitors/monthly_history_extra'] = array(
    'title' => 'Monthly history extra',
    'description' => 'View pages count per month.',
    'page callback' => 'visitors_monthly_history_extra',
    'access arguments' => array('access visitors'),
    'file' => 'visitors_extra.admin.inc',
    'type' => MENU_NORMAL_ITEM,
  );

  $items['visitors/monthly_history/graph_extra'] = array(
    'title' => 'Monthly history extra',
    'description' => 'View pages count per month.',
    'page callback' => 'graph_visitors_monthly_history_extra',
    'access arguments' => array('access visitors'),
    'file' => 'visitors_extra.admin.inc',
    'type' => MENU_CALLBACK,
  );
...
function visitors_monthly_history_extra() {
  $items_per_page = variable_get('visitors_lines_per_page', 10);

  $header = array(
    array('data' => t('#')),
    array('data' => t('Month'), 'field' => 'm', 'sort' => 'asc'),
    array('data' => t('Pages'), 'field' => 'count'),
  );

  $sql = sprintf('select count(*) as count,
                  %s as m,
                  %s as s
                  from {visitors}
                  where %s %s %s
                  group by m %s',
                  visitors_date_format_sql('visitors_date_time', '%Y%m'),
                  visitors_date_format_sql('MIN(visitors_date_time)', '%Y %M'),
                  visitors_date_filter_sql_condition(),
                  visitors_bot_filter(),
                  visitors_ref_filter(),
                  tablesort_sql($header)
                );
...
function visitors_bot_filter($text = 'visitors_user_agent'){
    $bots = array( '"bot"',
                   '"crawl"',
                   '"slurp"',
                   '"spider"',
                   '"wiki"',
                   '"curl"',
                   '"search"',
                   '"sexy"'
             );
    $lastbot = array_pop($bots);
    $botfilter = ' AND NOT (';
    if (!empty($bots)){
        foreach ($bots as $value){
            $botfilter .= ' LOCATE(' . $value . ', ' . $text .  ') > 0 OR';
        }
    }    
    $botfilter .= ' LOCATE(' . $lastbot . ', ' . $text .  ') > 0)';
    return ($botfilter);
}

function visitors_ref_filter($text = 'visitors_referer'){
    $refs = array( '"niki-boden"',
             );
    $lastref = array_pop($refs);
    $reffilter = ' AND NOT (';
    if (!empty($refs)){
        foreach ($refs as $value){
            $reffilter .= ' LOCATE(' . $value . ', ' . $text .  ') > 0 OR';
        }
    }    
    $reffilter .= ' LOCATE(' . $lastref . ', ' . $text .  ') > 0)';
    return ($reffilter);
}

You see, what I basically have done is the inclusion of the some filters. So it isn't necessary to hack the whole module. In your case I'd consider to extend the SQL or add a new statement with a GROUP clause and COUNT(*). The shortcoming of the Visitors module isn't the missing functionality, it is the lack of the programmers to implement the Drupal hook system (hook_db_rewrite_sql() ) properly.