I find vinegar to really not an effective cleaning tool. Yes, it is safe, but doesn't really kill a lot of bacteria so what's the point. Especially with dilution of water, you are affecting the pH which is essentially what you are trying to kill with.
I would use 10% bleach solution (1 part bleach/9parts water). First, I would clean the cage with soapy water first and scrub any debris off. Bleach really needs contact time to work, so once you spray it, it needs to sit for 10 minutes to allow to do it's thing.
Rinse well with water. If you can still smell bleach, rinse again.
Obviously, your cham will need to be somewhere else.
Peracetic acid is a good disinfecting substance. Can be made with off the shelf white vinegar and hydrogen peroxide. More reading on this subject must be done before application. http://en.wikipedia.org/wiki/Peracetic_acid