She's likely nursing early morning and/or late evening when you're asleep. And it only lasts a few minutes. The does knows it's time to nurse because her milk will be full and she will need to relieve the pressure. So if you put the kits with her in the early morning and late evening, then at one of those times or both times, she will be ready to nurse them. If she doesn't, then I would wait an hour and try again. She may not nurse both times. Some does will only nurse once a day. So you'll have to discover when your doe is wanting to nurse.
Yes, I would put him back with the others and keep an eye on all their wounds for signs of infection, which they would need to be seen by a vet if the wounds are deeper than the skin layer and/or the wounds get infected.
Or you can keep them with her and risk possible injury to the kits. It's not what I would do, but that's up to you if you want to take the risk and keep the kits with the doe.