I will never understand how a man could value sex so much above his relationship that he would use a prostitute behind his wife's back, I'm very glad you're not considering that! I can't believe people have suggested that as a solution. Seems to me it's a way to destroy a marriage, not build one. :/
Your situation is a tricky one by the sounds of it, it sounds as though your wife is very disconnected. Do you think that she could be suffering with some level of post natal depression? Did she enjoy physical contact and cuddling etc before getting pregnant?
I spend a lot of time on parenting forums these days and it's surprisingly common for women to suddenly 'go off' their partners during pregnancy, it's not something I've experienced but a lot of women on there say that they suddenly find themselves hating kisses and cuddles and feeling angry towards their partners for no particular reason, perhaps it's hormonal or something I have no idea. It's also something the pregnancy books touch on. Do you feel that this has happened with your wife?
If you're feeling like you're taking on too much with the housework and childcare then that's something you really need to discuss with her, and reach some kind of compromise, or you will end up resenting her (if I'm honest, it sounds a bit like you do already). We've decided that I will be a stay at home parent when our baby is born, but I've also made it clear that it's a full time job in itself, and I'll still need help with household chores and looking after the baby in the evening (especially as I have a chronic health condition), which my OH is more than happy to do as he's excited to get stuck in. If we didn't compromise, one of us would end up feeling spiteful in the end and it could affect our relationship. The joys of parenting, eh?!
When it comes to the sex, it sounds very frustrating, but right now it doesn't seem like your biggest problem. I think that by working on your relationship, being open and honest with each other, and learning how to co-parent and take care of the house fairly, you will open the door for sex further down the line. Perhaps I'm not fully seeing it from your POV as I'm a woman, so it's easier for me to understand how your wife might be feeling after pregnancy and birth, but I think this is a very normal and common thing for couples to go through after having children, and there shouldn't be a time pressure on women to be ready for regular sex again. It's so different from woman to woman, you might be seeing the fact that she gets more sleep than you and her birth seemed easy etc, but at the end of the day you have no idea what she feels like, her idea of being ready is probably very different to yours. Also, I wouldn't compare yourself to other couples and think "it's been X amount of time, things should be fine now", because again, it's so individual.
I'm not really sure what else to suggest as I'm not best qualified, but I really hope things improve for you both soon. :)