Buff Off Heavy Grime
Pick up some polishing compound at a home center or an auto parts store and use an auto buffer to polish off the offending scum. If you don't own a buffer, you can buy one for as little as $20 or borrow one from a gearhead friend. If possible, remove the doors and take them out to the garage to avoid messing up the bathroom.
Duster for the Vertically Challenged
Clean Grout with a Bleach Pen
For really mildewed grout, you may need a second application, and it can help to gently scrub the bleach into the grout with a toothbrush before allowing it to work for 10 minutes. Make sure to run the fan in the bathroom and to avoid skin . This method is best for light or white grout. If you have colored grout, test a small area first. It might fade.
Instead of trying to find a bleach pen at the store, the cleaning solution to grout might already be in the bathroom.