MeepyCat, I do understand that the OP's children are eventually going to learn that their dad cheated on their mom and that this fling resulted in a child who is their half-sibling, and I'm not necessarily saying that her children shouldn't find out until adulthood. I think that in a few more years, the OP's children will be better able to understand -- probably not why their dad made such a horrible choice, but why their mother doesn't really want to bring this child into the home and have a relationship with her.
I can also understand why the OP's husband would find it more convenient, for himself, to explain to the kids now, while they're still in at the age of being more likely to think, "Daddy did something bad, but now he's really, really sorry, so Mommy should just be nice and let our sister come to our house." In this scenario, the kids would have accepted the adultery at an early age and would already have a relationship with the child who is taking away some of their dad's financial and emotional resources, by the time that they understood more fully what a really awful thing their dad did to their mom and to them.
You're absolutely right that this affair will affect all of their lives forever, but I'm not sure how helpful it will be to the OP's children to hear about things like how the child support payment affects their parents' ability to provide certain things for them, or their ability to take as many vacations, or what have you.
I'm sorry about your husband's pain over what his dad did to his mom, and I'm sorry that the OP's children will also have to experience some pain because their dad cheated on their mom. Only the OP can decide whether she wants to stay in the marriage, or whether at some point she wants to bring the child into her home and make her part of daily discussions with her kids as to why Daddy doesn't have as much time or other resources for them. Honestly, if I were the other woman, I'd want to keep my child as far as I could from having to be in such a horrible position, but of course I'm not saying that her feelings should even come into this. So I'll go on to say that if I were the child, I'd rather not be dragged into this sort of a situation. I'd rather just be raised by my mom, and maybe a nice stepdad someday, and have some nice outings with, and financial support from, my bio dad.
I have just remembered a couple of situations I know of where a husband cheated and a child resulted from the cheating. In one case, the wife initially wanted nothing to do with the child who was born within a very short time of the birth of her own baby, who incidentally was born with syphilis because of her dad's philanderings with not just this woman but others. Later on, she started letting her husband bring the child home for visits, and they even tried at one point to get custody because the child's drug addict mother wasn't taking great care of him. They did eventually divorce, not just because of this, but because of the husband's chronic problems with dishonesty and unreliability.
In the other situation, I was friends with the wife for a couple of years but we lost touch after my family quit going to the same church. We met when her youngest daughter was nine, and I learned early on that this child had a sister who was the same age, whom my friend had learned of soon after giving birth to her daughter. She and her husband stayed together for a short time, but her husband was not at all apologetic and was actually kind of cocky about the whole thing, and she wasn't able to deal with his attitude, so they divorced and he married the other woman. The last time I talked with the mom, her older two kids didn't like visiting their dad anymore and she was no longer making them go, but her youngest actually had been wanting to go live with her dad, because she said it was more fun over there, and my friend had pretty much decided to just let her go. The ex wasn't so great about paying his child support but really treated his youngest like a princess and bought her lots of stuff and took her on fun outings when she was with them.
Last summer at a homeschooling park day, I saw a face that looked like this woman's oldest daughter's face, and asked if she was so and so. It turned out that both my friend's oldest and youngest daughters had gone to live with their dad and be homeschooled by their stepmom, with whom they seemed to have a good relationship. Of course, for all I know, my friend's ex may not have even been honest about having a wife and kids when they got involved. I think men who will lie to their own wives are quite liable to be lying to the women they cheat on their wives with.
That said, I think that this is a choice that only you, the OP, can make. No one else is in your exact same shoes with your exact same feet. Just do what feels best for yourself and your own children, and don't let your dh make you feel like you have to put your own feelings aside in order to make this tangle that he's created any easier for him to work through.