If the wood wheel and shelf are painted with plastikote it'll be fine to "wash" - maybe with a soaped sponge and then rinse off with a rinsed sponge - rather than completely immersing it. Just make sure it gets into every little bit and you don't miss a bit. And fine to spray with disinfectant afterwards.
If they aren't plastikoted - well you could still use soap and water on them and they'd probably be fine - maybe be a bit waterstained/water marks.
The wheel might be the thing that sustains most damage from water and disinfectant spray. I have washed a karlie wood wheel before (a few times) and scrubbed it in the sink with a brush! But that was without the cork. However with a different wood wheel when I did this, it started to fall apart (the wood veneer on top peeled off). The cork may also sustain some damage if not plastikoted but you could soap it and be gentle
TBH it might be easier to buy a new wheel. But then if it came to that you could have a go cleaning the old wheel and if it comes out fine you don't need to buy a new one!
Which cage/shelf is it? If it's somehting like the barney or alexander, the wood is quite think - the shelf would stand being scrubbed with soap and water though but it would take a few days to dry out. After that you could paint it with plastikote to seal it - so it would seal anything in (eg eggs that were missed) and make it easier to clean another time with some protection.
I have actually just found an old post of mine from 6 years ago
Where I researched how to kill mites. Freezing won't do it unless it's -70 degrees apparently! It needs to be oven baking and that was recommended at about 120 degrees c for a couple of hours - I think that could burn things . Some sites actually recommend chucking and replacing any wood items or things that can't be washed. That might be ok with something like a bendy bridge or house (a bark wood house would be very difficult to rid of mites by cleaning and the place most likely to have eggs or mites) but a bit unnecessary with a shelf.
So basically smooth wood items - given them a good scrub with soap and water. Really soap them, scrub, then rinse off - then dry and paint with plastikote. You could try baking bark wood items but probably safer to replace them.
If pet disinfectant isn't going to kill them (I'd have thought it wouldn't do them any good!) then I'm wondering if white vinegar might be better as a second clean. I know vinegar can help kill moth eggs. It's acidic obviously and according to google vinegar will kill just about any mite. The soap and water is to remove any invisible eggs or droppings.
Might also be worth asking your vet what they recommend using.
In terms of the hamster - giving immune boosting foods will help as well - to try and prevent a recurrence. So as well as the usual hamster mix, a pinch of hemp seeds every day will help. They are supposed to contain all known minerals and vitamins and are immune boosting. I used to put them on a separate little dish rather than in with the food where they disappear. They just lick them up and seem to like them. I get the Linwoods shelled hemp seeds (they're on Amazon or at Holland and Barrett).
Half a brazil nut once a week is an immune boosting treat as well. Brazil nuts contain something that is high in immune boosting. (Not in its shell obviously) and a nice treat for the hamster.
Linseeds are also brilliant for fur and helping it grow back and thick and good condition. A pinch of those on the same dish as the hemp seeds every day maybe.