Need help sorting “My Sites” Alphabetically

I have 40+ sites showing up in the “My Sites” page with more planned. Unfortunately WordPress doesn’t sort them alphabetically, and that makes it a pain to move from site to site during routine updates and maintenance. I’ve tried adding

asort($blogs);

to wp-admin/my-sites.php, but that doesn’t help either. And regardless, I’d rather do this using a filter in functions.php rather than modifying a core file. Making the issue even more complex is the fact that the list is split into four columns, and while a horizontal alphabetical ordering would be a huge improvement, vertical (by column) would be much, much better.

I’ve been searching for answers for this for a while and coming up empty, so any help would be appreciated. (My PHP is pretty basic, so spelling out the answer would definitely be appreciated.)