I use Poop Off for daily and weekly cleaning. For monthly cleaning, I use F10 SCXD, vet grade disinfectant. However I'm fortunate and can take Tico's cage outside to be hosed down. If I were if your shoes I would probably buy a hand held steamer for monthly cleaning.
I clean my cages with undiluted white vinegar.
Each morning I do a quick wipe down of any areas on the cage that are soiled, including cage bars, as well as any perches and toys.
I change the newspapers I put on top of the bottom grate, clean the food and water dishes, and use a little whisk broom to sweep out any debris in the pull out tray before giving it a wipe-down with white vinegar.
For more intense cage cleanings, I have a little scrub brush I use to get any debris out of the crevices at the bottom of the cage. It works well dry and to get off stuck-on debris I simply wet it with white vinegar.
When toys, perches and playgrounds need a more intense cleaning, I put them in the bathroom sink and scrub them thourghly with white vinegar and then let them air dry in the house before replacing them. I do this on an "as needed" basis -- not on any particular schedule.
I personally change out the cage liner and wipe out the tray with water and a paper towel as needed, generally every 1-3 days, depending on how many birds I have in the cage, and whether or not they're breeding (I keep breeding cages super clean to ensure no diseases can reach the breeding parents or babies). Every time I get new birds or move birds into a cage where I had other birds housed, I take the entire cage into the bathtub and spray it with vinegar, scrub it down completely with dish detergent, rinse, spray with vinegar again, and then rinse completely. I put mine outside to dry, but I think you could towel dry them inside if that isn't an option. I also do this complete disinifecting and scrubbing of each cage every 1-3 months, depending on how many birds are in it and how messy they are haha!
Miley (Yorkie), Kody (German Shepherd), Rain, Stormy, Dawn, and Dusty; RIP Sky, Breezey, Luna, Romeo, Juliet, Hermione, Eccleston (starling), Piper (starling), and Tulip (starling).
I clean all my birds cages with undiluted white vinegar also I don't think you need bleach, and the fumes would be harmful to the birds. Scented soaps could be harmful too.
I usually keep things clean around by vacuuming and cleaning the pull out poop tray. When the cage needs a proper clean, I pull it apart, take it outside, wipe/scrub everything with white vinegar, and then hose it down, or you could rinse in a sink for small cages.