Generally speaking (and based on Arizona law), child support terminates upon emancipation, which is basically age 18 or high school graduation, whichever comes later, but not past age 19. However, the order doesn't automatically change or stop; you need to file something with the court to modify the obligation. Regarding the remaining minor child, although he should not owe support to the mother (since the child is not in her care), both parents still have an obligation to help support the child. There is probably nothing you can do about the year she lived in your household; if a modification had been filed at that time, the court might have terminated his obligation to pay (except for the arrears) and possibly ordered the mother to pay, but since the court doesn't make retroactive orders, it may be too late.
One option to consider is asking the mother to give credit for the support that ways paid during the year the child was in your care (and apply that amount against the arrears), and for her to voluntarily terminate his obligation to her at this time. If she won't agree to do this, then the courts would need to be involved.
** This response is provided as general information only and does not constitute legal advice or otherwise establish an attorney client relationship. **