I recently had my 4th child. I have children ages 14, 11, & 9. I was on bed rest for 7 weeks and then had a c-section. I was not able to be a mom for about 3 months, so you can only imagine what my children were going through, they had to be the mom along with my wonderful husband. So, by the time I was back on my feet again, my older children were falling apart. What I started doing, is taking one day a week to dedicate to my other children. The way it works is they each get 1 Thursday a month for themselves (1 child alone with me), and then the last Thursday of the month or the 4th week is for all of them together. My husband stays home with the baby and his son when I do this. I let my children chose what they want to do, as long as it is not a $200 shopping spree. So, for you being a single mom, (I was a single mom of 3 for 3 years so I know where you're at) you have to find someone, a family member, a friend or just a babysitter you can trust to take care of the little one so you can give your older daughter what she needs, a date.
And I know you stated that you didn't want to take time from the other child (the 2 yr old), but what you did originally was take time from that older child to "have" the newest child. You aren't going to hurt your younger child by taking time out for your oldest, you love them both so you have to give to them both, each individually and together. You can do it, trust me I know what it is like to juggle 4 plus children, because I have step children.... this baby we had made 8 for us, so give it a go and you'll see a difference in your daughter. Try to take time with her as well when the baby sleeps, it is soooo important, especially now that she is coming into puberty!