Aww, I'm sorry about the hen you lost. But I don't know if hen-hatched & raised chicks would be incorporated into the flock as adults any easier than brooder-raised ones. Also, if you hatch chicks you will probably have more roosters than you need, and will have to plan for that. Chicks take 18-20 weeks to mature, and most hens won't stay with them for that long. The hen is usually done with them after 4-8 weeks or so, and then the chicks still need to be kept separate for a while after that.
If you want more hens, you could buy sexed pullet chicks from a hatchery or feed store. Brood them yourself, and when they're about 10 weeks or so they can be given time to free-range with your adults, and start to get acquainted. They could also be housed next to each other to allow them more time to familiarize themselves with each other.
There's not a lot you can to do encourage broodiness, it's up to the hen and the voices only she hears in her head. Leaving eggs in the nest may encourage egg-eating, breakage, or other critters to the coop. If a hen feels broody she'll set on anything or nothing, golf balls, wooden eggs, an empty nest. It's more up to the individual temperment of the hen and her hormones that stimulate broodiness. Once a hen has started to go broody, there's a lot you can do to encourage that by keeping her in a quiet secluded place of her own.