Over-Optimized Website

I'm looking for advice for what you would start with if you were working on a website that was extremely over-optimized for 1 keyword. So, for example, I'm going to pretend this client is a dog trainer in Toronto (I can't publicly post the URL).

I've read places that having exact-match anchor text links to inner pages in the footer of the site can cause problems and removing it has resulted in big ranking jumps. I'm looking to see if there are other big items that you would tackle first if this was your client.

Some examples of things the site has:

- There is a page for dog training under their Services menu. However, internal links on their site link "dog training" to both the homepage and to this service page. Is that going to cause issues?

- The anchor text for internal linking is almost always the exact same word - "Dog Training".

- There is a banner that goes across the top of the site that appears on every page that says "Dog Training Toronto". I'm guessing I should remove that. Would the same keyword being overly used on every page cause confusion?

- Almost every image on his site is saved in the format "Dog Training Toronto".

I'm looking to see if anyone has general tips on where to start with a site that has been over-optimized for 1 keyword. He actually has a ton of good content on his blog that gets a ton of traffic (because it's actually useful) so it's not that his content sucks - it's just been overly structured and SEO'd to death.

I found a few articles on this but other than the footer advice I didn't find too many case studies of others that have run into this issue and done a few steps that actually worked.