Oh dear, I feel for you. I can tell you with certainty that you will not win an Infant Class Size appeal on the grounds that you need a school with after school care and her siblings attend there. You just won't. (I used to sit on admission appeals panels, so have knowledge of them).
Most LAs have posters in Doctors' surgeries, libraries, school receptions, preschools, newspapers, etc. Some LAs send letters, but it's becoming more unusual. It isn't done in my LA, which is one of the largest LAs in the country, as it's just too expensive. Unfortunately, appealing on the grounds that you didn't know you should have applied won't succeed because everyone else managed to apply.
What you can do, is make sure that the PAN of your preferred school is a multiple of 15. If it is 15, 30, 45, 60, or 90, then the appeal would be Infant Class Size. Check the rejection letter to see if it says that she can't have a place because they are full and the Infant Class Size Regulations restrict class sizes to 30 children. If so, your grounds for appeal are extremely limited.
If the PAN is a different number, not a multiple of 15 and it was not ICS, then you have a more flexible prejudice appeal process, where you would just have to show that your DD needs the school place more than it would cause a problem for the school to have her.