My cleaning method is this :
1. Remove gecko, one hide and some substrate, place all of these in a bucket (with sides deep enough that the leo can't climb out)
2. Remove (and unplug) all items from within the viv
3. Use a steam cleaner on the inside of the viv, ensuring that everywhere is steamed and wiped
4. I use a loose substrate, so, add new substrate
5. Use a steam cleaner on the hides etc., ensuring that everywhere is steamed and wiped, replacing inside the viv as you go
6. Plug in all heating etc.
7. Return leo to the viv
8. Steam clean and wipe the remaining hide
9. Replace the last hide
10. Top up food and water bowls and moist hide substrate
Steam cleaners can be purchased relatively cheaply, and this way you're only using water to clean.
Remember that you need to wash your hands between handling dirty and cleaned items (otherwise you might as well not bother cleaning, as you'll just transfer the dirt and germs).
I'd be tempted to use more than "just water" on the reptile carpet, as unless it's really hot, you won't kill the bugs.
Depending on your ambient temperature, you may want to add a hot water bottle underneath your bucket to help keep your leo warm.