Something similar to this happened in my family.
I want you to know that am not here to judge you.
In this case my uncle had a daughter and he didn't all the rest of his family about her. we are a close family and some of us knew about her and some of us didn't.
it was painful for all of us. the ones of us who knew were being forced to keep secrets from the those who didn't. we all worried about their reactions when they eventually found out and whether they'd be angry with us even though we were only children who had been told to keep our other cousin a secret.
eventually the cousins who didn't know found out - one of the cousins who did know told them a couple of years ago. The cousins who didn't know were hurt with us, even though they loved us and didn't understand why they had never been told. it was actually all their family unit who were not told ie their parents too and my uncle's sibling, who he typically got on with very well. they were all understandably distressed and bewildered.
I have no advice for you, OP. I think you are in a very difficult position and I have a lot of sympathy for you. I think that other posters are being unkind to you but that the situation is much more nuanced than they like to admit. just giving you a massive kicking is so unhelpful - and so much the easy option.
I know exactly what keeping a secret like this is like - as I was forced to keep my uncle's secret for so long. it makes one terribly anxious.
I don't hate my uncle at all even though he was the one who put us through this. I have spoken to him about it and, if i'm honest, he hasn't really acknowledged what it cost us (the anxiety and the lying, and the fear of losing our cousins' love) or could have cost us (by which I mean my own close relationship with the cousins who didn't know, if they had chosen to be angry with us for keeping the secret too). He would say he had no choice, even though he obviously did. i don't think he particularly feels guilty and sometimes that makes me feel quite angry with him!
What I acknowledge about my uncle is this: that he really got himself into a pickle and the longer he left it, the more impossible it became to tell the other cousins. the reason he didn't tell them, I think, is because our cousin was illegitimate and the uncle he didn't tell had mh problems. He was worried that the uncle was too vulnerable to be able to keep the secret from our granny and then she'd find out and she'd be devastated. but when she died he just didn't tell my uncle and it kept dragging on. I still really love him though, and of all the uncles he is my favourite. But we have always enjoyed a good relationship and I was the only cousin to confront him after the news came out. his brother never did - they have to my knowledge never discussed it (mind boggling) and that sort of makes me think that maybe not telling him while granny was alive wasn't the wrong thing.
My uncle wasn't married to anyone, so didn't cheat like you did. he was in a much less complicated position than you. he always acknowledged his daughter and paid maintenance for her and was a signficant part of her life. they have a good relationship. he had no other children so much less complicated than you.
From my cousin's point of view i don't think it was great for her to grow up knowing she had cousins who didn't know about her. it was damaging to her, although she is a wonderful person and was a fantastic child. both she and the cousins she didn't know have missed out hugely knowing each other and that is something which can't be replaced. they love each other now but aren't particularly close because they all have their own lives and they were adults when they met.
fortunately my own anxiety about what woudl happen to my cousins relationship with me was misplaced and we are still as close as ever. they also still love the uncle who kept the secret from them, although they are saddened and puzzled by it. it's something that we all talk about together now, and I mean all of us including the cousin who was not known.
from my point of view I am less concerned about your wife, although obviously I don't envy her. Like you, i'd be so worried about both your daughters not knowing each other. I have no advice because I feel that the arguments are actually very powerful both ways. If you continue to keep the secret whilst you are alive, then yes both your wife and your daughter will really struggle with your death - it will be so much the worse for them. they'll never get the chance to process it with you, which is what they deserve. and, most importantly to me, your daughters may never get to know each other in a meaningful way. but if you tell them and you really feel that there is a risk that they may simply not wish to know the little girl, then what is the point? it sounds like your marriage is not emotionally intimate, and again I make no judgement, because there are loads out there which are relationships of convenience and suit each party just fine. i too am struck by your wife saying that she'd rather not know if you were having an affair - that, for me at least, is very telling.
For one, I have been impressed with your replies so far - because it strikes me that you are not trying to shirk your responsibility but face up to it and do the right thing, whatever that is. that's certainly more than my uncle was capable of. this can only bode well if you do chose to tell them.
I do not think that you are a coward because it takes real strength to carry on as you have done with this dreadful secret (albeit of your own making) and I think that you are genuinely worried about hurting your wife and daughter and want to protect them for that.
If you do chose not to tell them (and I'm inclined to think I wouldn't) I'd leave a letter for them with your will explaining your reasoning. It's not fair and it's not really good enough, but I don't think it's ok just to leave them with the will and no explanation.
the only thing I do feel strongly about is that under no circumstances should you cut your little daughter off. she may have some hard questions for you in the future but she needs to know that her daddy loves her and it sounds like she loves and values you too. My cousin and my uncle have benefited hugely from having each other in their lives. I think your younger daughter needs to know that you are not ashamed of her and that she is valued by her daddy.
Like your older daughter, I am also a feminist.