First and foremost I will test the symptomatic browser issues across a range of browsers to confirm the diagnosis - I have several virtual machines installed locally, these range from ie6 to ie9, I also have a windows machine with the latest ie version to test - I will be able to test the site across each browser before during and after any fixes are implemented, to ensure existing compatibility is retained.
I can also double check the site works in the other modern browsers (firefox, chrome, safari, opera, on mac, iOS and android).
This is just to ensure that the problem is correctly diagnosed, before committing to a fix which could open a can of worms.
If it is a case of some visual errors in ie8 and ie9 only, then I expect to identify and fix them certainly within a day.