I'm sorry about the situation. However:

1. You need to clear up whether or not Delta offers child fares and how to obtain the fare (it seems there is some debate above). It doesn't matter what KLM offers, as you bought the tickets through Delta with Delta flight numbers.

2. If Delta does indeed offer child fares, you can try longshot - you can ask Delta for the fare difference as a credit. If you did this already when you called them, then ignore this. However, if you asked for a refund, that is the problem. Delta is more flexible than other airlines, but more with credits than refunds. When change fees were a thing, I had over $400 in change fees waived.

3. IF Delta does indeed offer child fares, and #2 does not work, your only recourse would be to cancel your current booking and rebook with the child fare. If the fare decreases, your children will receive a credit (NOTE: you will be rebooking at the current fare, which may NOT be the same fare you originally booked. You will need to see what the current fares are).

4. IF Delta does NOT offer child fares, unfortunately you are out of luck. When booking airfares, it's the fare rules of the ticketing airline that are followed.

5. Finally, DO NOT take what the KLM agent told you about Delta getting the discount as the truth. There are complex calculations of how fares are distributed between joint venture partners.

Hopefully you can get some resolution to this.