Hi there. I would wash anything washable (like plastic or ceramic) in washing up liquid and water. Things that aren't washable could be baked or go in the freezer. Personally I wouldn't keep bendy bridges as they have so many crevices. What a total pain!
The trouble with putting things in the freezer is he doesn't have them while you're waiting. Baking might be quicker but he is still going to go a day or two without his familiar things.
I would bin all the bedding and nest from the cage, clean it out with this disinfectant (it's the one our vet recommended and sold by local pet shop). It doesn't smell too strong, but it's still a good idea to rinse it and let it air a bit - I put the whole cage base and top in the bath and used the shower to rinse it.
https://www.amazon.co.uk/Beaphar-Dee...e+disinfectant
If the wood shelves aren't plasticoted I wouldn't use the disinfectant on those, but wipe them down with a damp cloth over the sink and then either bake or freeze them along with other wood items.
I had to do all this after a moth infestation (ugh) from food (I always freeze the food now). After that I tended to stick to smooth wood items, ceramic and plastic as much as possible as they're easier to clean!
I put everything in the freezer for a week. I believe baking is quicker and simpler but don't have experience of that - someone else might.
If you went the freezing option, you could set the cage up after cleaning, with new bedding and a cardboard house and a few toys that have been cleaned (eg plastic or ceramic) and just add the others when they're ready. It's a bit of an upheaval for him but as long as everything is in the same place I'm sure he'll adjust - especially if you add a couple of treat sticks or a new toy maybe. He might need a week or two to settle in again and avoid handling for 2 or 3 days again, but he will probably show if he wants to be left alone or come out. I'd definitely leave him alone for a day or two and just talk to him through the bars.
What bedding are you using? Is it stored ok? I think if it gets damp it can be an issue. Assuming it's a big bag of Fitch and not damp, just stick a whole load of new Fitch in
I believe some people put dismantled wheels through the dishwasher
I'd personally wash the wheel, food bowl and water bottle by hand and disinfect the water bottle and food bowl with boiling water from the kettle.
Sorry you're having all this - it really is a pain for you and Quigley. But once it's done it's done and just moving forwards and settling in again.