I am not a breeder, but I also had a chinchilla that I didn't know was pregnant. Here are some basic guide lines that I used and what others have already said.
Remove the male from the cage. The female can get pregnant again after giving birth.
Bar spacing cannot be more than 1/2 inch All around. Kits can climb and will squeeze out of any opening that they can get by.
If you cannot find a cage with that spacing, use wire mesh. I bought a roll from Home Depot and tied it to the bars of the cage with thick wire. The mesh cannot be under 16 gauge because chinchilla can chew threw it. If you do this, make sure there are no sharp edges sticking in the cage or where a chinchilla could potentially hurt themselves. I used grid a grid type wire.
No levels in the cage. Kits do not have depth perception when young and will fall. Also, the mother can accidentally jump on her kit and injure it.
Oxbow pellets and timothy are ideal. I bought alfalfa hay as well to make sure the mom was getting good amounts of calcium. You do not need as much alfalfa hay as timothy. No treats! Always have a good supply of fresh water. The mom needs a lot of fluids.
Buy a digital scale that weighs ounces and grams. Weighing the kit(s) is important, and so is weighing the mom. Keep a record of this every day. You may want to start to weigh the mom now and see what she weighs daily. If possible, put the scale in the cage and see if she will sit on it. Try not to handle her.
If you have Carefresh bedding or anything fluffy, I recommend taking that out and switching to aspen shavings. The kit I had starting eating the bedding and was not nursing as much to get essential nutrients.
I haven't looked to much into this, but if the mom does have multiple kits make sure she is able to care for them. Sometimes there is a weaker chinchilla and other kits may nip at him or not let him nurse. You may need to rotate kits to help the mom when nursing. She may not be able to feed them all at once, and as I said, there may be one that may not be getting as much. I would look more into this yourself.
Remove bedding that has anything from the birth on it.
Make sure she has a quiet space that is not too bright. Keeping her stress free is important. Try not to handle her too much.
If all goes smoothly, you will mainly just be there to check up and provide for the mom. The mom takes care of the kits for the most part. Of course, keep looking into this and try to contact a local breeder for advice (that's what I did).
I am NOT a breeder. I am simply going off of the things I did when I found a kit in my chinchillas cage one day. Hopefully someone who raises chinchillas and breeds them will respond and help more.