When you add eggs later, those eggs are likely to hatch later - and that means mommy is likely to have left the nest to care for the chicks that already hatched, before the last eggs hatch, and so they might die in their shell just one day short of hatching, provided that any of the first 4 eggs hatch. So if you have an incubator, I'd keep that ready when hatch day approaches, and then I'd watch the hen very closely to be able to move the eggs to the incubator as soon as she gives up on them. If she already has chicks and you hatch more in the incubator, she might be willing to take are of the new chicks as well if you place them with her once they are dry and running about.
The 'typical' successful hatch setup doesn't have a wire bottom. I don't have coturnix myself, but I've been reading this forum for months and in that time there has been several posts about broody coturnix hens, most of which have been housed in large, uncrowded, not wire bottom cages/aviaries. If your hen only had two cage mates, I guess the uncrowded thing matches, but the wire bottom is unusual.
I's a good thing you were able to get the others out without causing her to leave the nest.
When you say you want to take the chicks in and keep them in a brooder under lights, does that mean you don't intend to let the hen try to raise them? I think you should give her a chance, but I would probably make another cage for her and the chicks - one that doesn't have wire on the bottom and the first ~10 cm. of the sides. But it will be risky to have her hatch the chicks in the cage she is in, the chicks could fall through the wire and die before you even know they hatched. Or they might just walk on top of it and look like I'm a complete idiot for worrying about it
I'm not convinced that newspaper on top of the wire would fix the problem, the mother could cause the newspaper to move so they could still fall through the wire, or they could walk through the wire on the sides and not the bottom. Or they could get caught between the wire and the newspaper.. But again, maybe I'm just worrying too much ^^