There is a real world mammal that has a queen and caste system similar to what you describe: the naked mole rat (Heterocephalus glaber). BBC wildlife videos about naked mole rats
To turn a human woman into a naked mole rat queen who can have big litters of babies, you need to sort out the human pelvis and/or the human baby. And possibly the human placenta.
Pelvis: birth is difficult and prolonged because of trade-offs between walking bipedally and having a big brained baby. If your queens are practically immobile, and just lie around all day (like a termite queen or naked mole rat queen), then the pelvis architecture can be changed drastically. Wide enough to squirt out the babies without much effort.
Human baby: alternatively, make human babies much smaller and even less well developed than they are now. So instead of an 8 lb baby, the queen has several kitten sized babies (8 x 1 lbs), for the same 9 month pregnancy. The 'factory settings' for the mammals is 8 nipples, and some beasties have 16, so the queen can have multiple babies nursing at once. Or, she passes them over to the infertile females to breast-feed.
Placenta: The human placenta is directly bathed in the mother's blood. This is very efficient in getting nutrients out of mum's blood and into baby's blood (only 3 layers of tissue to pass through). The downside is if something goes wrong, mum can bleed to death. If mum is producing 8 to 16 babies at once, I guess that risk goes up as she has 8 to 16 placentas? So perhaps the human placenta could be re-engineered back to the 'primitive' mammalian state of 6 layers of tissue:
baby's blood - placenta 1 - placenta 2 - placenta 3 - mum 1 - mum 2 - mum 3 - mum's blood.
Much less efficient, so development of the foetuses would be slower.
Meanwhile, menopause. Menoapuse isn't just about egg number, it is about nutrition and how much hormones you have squirreled away in your body fat. The average age for British women hitting menopause is 51 years old whereas subsistence level farmers the average may be as low as 42. So if you feed your queens a high fat diet, they can have a longer reproductive lifespan. (Possibly. Some of this delayed menopause is an ability to still having periods, even though you've run out of eggs.)
How many babies they'll need to have will depend on mortality in your society. If 50% of kids die of childhood diseases or get eaten by bears, the birth rate will need to be high to keep the population stable.