I use a bleach, water and anti-bacterial soap solution. Generally I just dry it myself with paper towels and redecorate. It would depend though on how much bleach your using, I tend to stick with a 1:10 bleach to water ratio. You shouldn't be overwhelmed by the smell of bleach, it should be hardly noticable.
Bleach when exposed to air will completely evaporate withing 24 hours, so if you really want to take the safe route you can go with that .
I always use a bleach and water solution for all my animals (just not on absorbant toys such as wood, ropes etc) and i've never had any problems. I don't use it every time i clean out the cages though, only when i'm doing a deep clean. Virkon is a really good pet-safe disinfectant - i use for all the absorbant stuff, you can get it through vets and some animal product suppliers in the UK.
for deep cleanings i have recently discovered an absolutely wonderful thing called the do it yourself car wash! the high pressure blasts ANY stuck on or hard to get dirt and the rinse, of equally wonderful high pressure gets rid of any deterigent smell. the hamster ball i tie up for a hidy hole has NEVER been cleaner. heck, i don't think it was this clean when i bought it. i love the car wash. and the BEST part, it takes only five mintues to do a DEEP clean. normally it will take me anywhere from an hour or more to deep clean by hand and not with nearly as satisfying results. i'll say it again, i LOVE the car wash. mind you, i only do the car wash for hard surface toys and the cage and i don't use wooden toys.
but i've never used bleach either. when i did the cleaning by hand i used a vinegar and water solution instead. the acid in the vinegar eats into any dried pee and disinfects and if i don't rinse it well or happen to make it a bit stronger i know that my rats are still safe though they may be a bit more wary to chew the cage bars if there's still a bit of vinegar residue left... but they won't get sick from it certainly.