I've got several sites in Joomla and I update them all to the latest software at the same time. One of my sites has K2 installed on it (thanks to a packaged template that I purchased). Unfortunately I can't find where the heck to update Joomla itself. I've updated the K2 extension, but I can't find Joomla. I'm several versions behind and getting very concerned about security. This seems simple, and I've looked on several occasions, but the control panel doesn't show a "maintenance" section which is where I find it on my other pages.

You have it backwards.
K2 is a Joomla! extension which means you update K2 through Joomla! not the other way round.

docs.joomla.org/J3.x:Updating_from_an_existing_version

No, I have it correctly. I just want to update Joomla to the latest version. When I try from the Control Panel, there is no option for Maintenance/Updates. If I go to Components and try to update I get an error (0 No HTTP response received). Am I just screwed? I feel like something went wrong with an install, but since K2 changes around the Control Panel, I want to blame it (which might not be the case). I did not have any problems updating K2.

Joomla! is the CMS, K2 is the extension. Updating the CMS has nothing to do with extensions.

Please visit the link I sent you in my previous post.

I did go to your link, and appreciate that because it gave me a second place to try to find how to update Joomla (had just been going to the Control Panel). Unfortunately it errors out (0 No HTTP response received), and I don't know what to do about it. If you have suggestions on that, I'd appreciate it. I don't 'think' it's a K2 problem now.

Since this is a core Joomla! issue, you need to address this issue to Joomla!'s forums.