How to diagnose a search engine ranking drop?
Some possibilities:
The sites who moved up in the rankings may have gotten quality links that boosted their relevancy (as opposed to you losing relevancy).
You may have lost quality links
The sites that leaped ahead of you made big improvements in their SEO and now are reaping the rewards
There was a small algorithm change that played to the other sites' strengths
There was a small algorithm change that played against your weaknesses
Also keep in mind that rankings naturally fluctuate. It's very possible you will return to your previous rankings soon without having to change anything on your site.
One thing you should do is check to see if any of the sites that moved ahead of you are new or if they made any changes (if you don't check out the competition regularly you may not be able to do this). If so, you may want to make similar changes and see if it helps you. Be prepared to roll the changes back if they do not help or hurt your rankings.
My advice whenever someone has this occur is always don't panic and don't change anything. This is a perfect opportunity for you to make changes that you really don't need to make and they end up hurting you in the end.
If your content is still strong then wait it out. Keep getting links to your pages. Don't panic. Be patient.
update
To diagnose a drop you need to:
Undo any recent changes and wait to see if your rankings revert back.
Make sure you are up to date on Google's latest changes to their algorithm and see if any of them apply to you. If so, make changes that should affect that algorithm change and wait to see what happens. If nothing changes then rollback your content as that was the last known well-ranking format for your content.
That's about it. If you didn't change anything and Google hasn't made any announced changes then you have to assume your competitors got better either through better content/SEO or minor tweaks at Google that were unannounced and difficult to test for. In this case just keep plugging away. If your content is strong, you will regain your old positioning at some point.
If you're using a good analytics program, check if the majority of your pages have taken a hit or just the top pages. Example: look at your middle of the road pages and see if they've also been reduced by 20 - 60%. If you see a consistent drop across your pages then you've likely been bitten by Panda and there's a long road ahead. Don't panic though, but there is no easy fix to the problem.
If you're just seeing changes in a top few pages then you really have nothing to worry about and should just continue what you normally do.
Given the age of your site, it's also possible that a number of incoming links to your site were devalued during an algorithm update and that caused a domino effect across your pages. So it might not have been directly related to your site, but more a reflection of the relevance of sites linking to you.
What's your site's main source of revenue?
You mention that the Google Analytics code now appears in the headers rather than at the end of the page; has that changed the page load time appreciably? The GA code may now be executed before the page content loads, slowing down the time it takes for Google to see that content; if you were close to some limits before, it's possible that could have pushed into a penalty area. OTOH, I'd not expect a page speed penalty to be anywhere near as drastic as yours seems to be.
You can check loading patterns with something like http://loads.in/ (and probably Google Analytics somewhere...).