One of mine had a broken leg a year or two ago and I had to have the cast changed a minimum of twice a week. No matter how you slice it, casting a chinchilla is high risk and incredibly draining on all parties involved.
I was advised against the cone because they can't reach around the cone to eat. If you choose to go the cone route, I strongly recommend picking up a digital gram scale to monitor any weight fluctuations. Weigh at the same time 2 days in a row. If he drops weight, he's not eating, and the cone needs to go.
About two weeks after the break, mine developed an open wound of some sort on the broken leg and started chewing furiously at the cast to get to it. She was getting Metacam from the get-go, but once they found the wound, they added antibiotics to the mix. Since yours already chewed his leg, you're dealing with the same risk of infection that I faced, so you'll want to make sure the vet keeps a close eye on that in case antibiotics are required. Personally, I'd go in twice a week for the first 2 weeks to have them check the wound. Better safe than sorry.
If you have to give antibiotics, you'll also want to give probiotics to repopulate the good bacteria in the gut that the antibiotics are killing. To further exacerbate the situation, some chins will go off their feed while on antibiotics, which results in the need to handfeed. For that, you'll need Critical Care or Essentials for Life.
The bottom line is that no matter what you do, he's probably going to find a way to chew through that cast. If he ingests any of the material, you're looking at a potential impaction or blockage. The only way I found to stay on top of it was to check on mine every 2-4 hours around the clock and clip off any frayed pieces as you see them. Casts only have so many layers, so if you see the stuffing or notice how thin the layers are, you NEED to get him into the vet immediately to get recasted. If you notice a problem off hours, you'll have to stay awake in the middle of the night to offer him food and sticks as diversions so that he doesn't chew the cast.
As everyone else stated, mobility needs to be severely limited to prevent further injury. No shelves. No wheel. No dust for a bit. Only food, water, hay and sticks to chew on. I gave mine dust showers by sprinkling it on her head in a separate cage after about a week. The issue is that if dust gets under the cast, it can further irritate the open wound and lead to an infection, so check with your vet on whether dusting in moderation is okay or not.
He's going to be completely dependent on you for everything since you'll be taking away his entertainment, so make sure you give him plenty of extra attention and as many sticks and toys as he wants.
I hope that helps a bit and I sincerely hope he heals quickly for you. It's a difficult position to be in.
Best of luck to you both!