What chinchilla food are you using? The mixed foods that contain fruits and nuts are very fatty. Consider switching to Mazuri Chinchilla Food. Chinchillas aren't usually big wheel users, but some will use the flying saucers.
For the most part, the excess weight is probably caused by the food and treats that you're offering. Switch to a healthy pelleted diet and minimize the treats that you're offering. Carrots break down into sugars, so don't offer them often, and don't offer a lot. You want MINIMAL treats, and fresh produce are treats.
As for the cage, make sure that it has many ledges and stories that the chin can jump and run around with. And, make sure to offer plenty of outside cage time.
If you only have one chinchilla, this is another potential problem, as they are very social creatures. Consider getting another chinchilla. Quarantine is important, and a slow introduction is necessary if you plan on keeping them together. Also if you get another one, consider a larger cage, depending on what you're using now. Having a friend will provide social interation and more playtime.
Mazes, balls, and other strenuous toys are not always the best becuase chinchillas overheat easily. If you opt for these options, keep this type of play at a minimum of maybe 10-15 minutes at a time.