I would just take the carpet out of the cage, hang it up outside on some type of clothesline and squirt on a mild liquid soap. Then using a power nossel on your water hose I would rinse to clear all the soap out. Leave hanging until dry.
For many years I used to use large bath towels in our boys cage, thinking this was the best bedding for them.... We changed their cage every night, so you can imagine all the towels we had to wash
Then I learned about CareFresh from this group, and I am thanking my lucky stars. It's easy, doesn't smell, and the cavies love to run through it. I can get by with not cleaning the cage for over a week. And no more stinky cage. That is a HUGE improvement from what we were having to do before.
I put a layer of the *non toxic* Aspen Wood chips down, and then top the chips with a bag of CareFresh.
You might want to think about trying CareFresh.
(Personally I think it's the greatest, as does our boys. They make paths through it, and brumes.)