I'm so sorry to hear that. I too think it's really odd that it would be positive. In the event that it was I went ahead and posted my experience with my cat. He was always sick with one URI or another or something else because the leukemia weakens their immune system. I chose the nurse it back to health and try everything option - it died anyway but own it's own. I didn't have it put down. That's a hard decision to make and I wouldn't want to be in your shoes, especially with kids. You can always try nursing it back to health and see if it recovers from the current sickness which if it doesn't at least you won't have to have it put down which might be harder on the kids than "Hey, we'll take her home and pray for the best and do all we can." Our vet said many cats can go on and live with the leukemia but they are always suseptible to infections because of the weakened immune system.
At the time I did not have any other cats to be concerned over spreading the disease - that is an issue because the vaccine is NOT 100% effective and even if the other cat is negative and is always current on the vaccine, living with an infected cat, sharing food and litter boxes, will always put it at risk.
So sorry to hear.