Okay...I add links to my blogroll. I choose a category for them. On my sidebar it shows as ONE list of links and is not categorized in any other way than within my dashboard organization. I don't know how to make it clearer than that.
Okay, the problem is that up until now, you had not made it that clear. Furthermore, you act as if we could see what you're talking about, when you had not posted the address of your blog until your last post.
All we have to go on is what you tell us. If you don't say it, we don't know it.
As for your particular issue, the problem is indeed the use of the get_links() code. You could probably replace that with just
wp_list_bookmarks(); all by itself and see an immediate improvement.
What you're going to do is to look at your theme's "sidebar.php" file. You can get it via FTP or whatever. Ignore the style.css, you're not going to modify that.
In the sidebar.php, you'll likely find something like this:
<?php get_links('-1', '...
Replace that code with just this:
<?php wp_list_bookmarks(); ?>
Yes, get rid of the Inspiration. Why? Because the wp_list_bookmarks() is going to do that for you.
Save it, and FTP the updated sidebar.php file back to the site, where you found it.
Now, when you load your page, you'll get something new. Instead of "Inspiration", you'll get "Blogroll" and your other link categories in separate chunks. You can rename the "Blogroll" or whatever the names of the categories are on the Manage->Categories admin page. You can add new Link categories, and they'll all appear too (as long as they have links in them anyway).