Formatting my archives sidebar in a specific way

Hi,

Firstly, I am trying to format my Archives sidebar in a certain way, so that it takes the format of the one in this example:

http://bldgblog.blogspot.co.uk/2013_03_01_bldgblog_archive.html

The second thing it must do is show only the months where posts from a certain category- blog (as I use posts for other purposes in the site) exist.

I did achieve the appearance using the following code:

<?php
$year_prev = null;
	$months = $wpdb->get_results(
		"SELECT DISTINCT MONTH( post_date ) AS month ,
		YEAR( post_date ) AS year,
		COUNT( id ) as post_count FROM $wpdb->posts
		WHERE post_status = 'publish' and post_date <= now( )
		and post_type = 'post'
		GROUP BY month , year
		ORDER BY post_date DESC"
						);
	foreach($months as $month) :
		$year_current = $month->year;
		if ($year_current != $year_prev){
			if ($year_prev != null){?>
			</ul>
			<?php } ?>
		<h4><?php echo $month->year; ?></h4>
		<ul class="archive-list">
		<?php } ?>

At this point however, months appear where I have no posts from the category – blog, so that when you click on them, no posts apear , whch is not ideal.

Any help would be gratefully recieved!