wow if you think that's dirty you wouldn't want to see my garage floor... LOL.
I grew up in Wisconsin though so I get what your going through right now in the winter time! It's even worse with mud and snow and dirty snow and nasty stuff on your car!
Looks like you have done a petty darn good job considering. I would try the easy not so expensive way first. Pressure washer and some bleach sounds easy enough unless that would hurt the floor? I don't know....
Course if your wife wants it a different color all together well then your in trouble!! Like the guys said above paint just wont last. I have been told myself epoxy is the way to go on a garage floor.
Good Luck
RC