If there is something in the contract that will affect the client negatively or if the old firm is suspected of being likely to cause problems then it may be that the clients best interest is suited by employing a reputable SEO firm to work on link building to strengthen the link profile before cutting ties with the old firm.

There will obviosuly be additional cost by following this approach however this would need to be weighed up against the possible rankings fall / traffic fall that might occur if the contract doesn't protect the client from the sort of situation described.

The way to rank for spelling errors is to optimise for them. Rather than littering the site with poor spelling which will not look good to the user you should try to include the misspelling as genuine content.

For example a light hearted blog post with incorrect spelling vs correct spelling in the title and page name. In the content t could mention the spelling mistakes and unusual terms that people use to find your site (gathered from the analytics data).

It may not be the most interesting post for the visitor and will struggle to gain page specific links but you will have a page optimised in terms of content for the misspelling. If you don't really want people to read it release it to the site at a quiet time and then follow up with a stong post which people will read and the spelling blog post will be overshadowed.

Hope this helps.

You could perfectly optimise your site on-page and still rank below competitiors due to them having a stronger link profile.

As mentioned in the other responses I would hope to see results inside of a month BUT it may be that you have to work for a number of months to catch up to where competitors are depending on how string their link profile is.