Hi, Wonder woman, you!
DS2 was born when DS1 was 3. We got our first au pair to help with the situation as I was very ill after his birth, and DS1 was extremely jelous and playing up a lot. He was even worse when the au pair came, he did not want to be with somebody else, he wanted mum, to be babied, and loved. I was very angry with him and could initially not understand his tantrums and his naughtyness, my husband gave him plenty of love and attention and said this was the way to go. (I am glad for your DD that your DH is doing the same)
I realized that I had to let the au pair handle the baby, DS2 was so young it did not really matter to him, but it mattere to DS1. So I would give DS1 some proper one on one, lots of love, attention and encouragement, while au pair was with baby. It took a few months, and then it eased off.
I hear you when you say you cannot afford a private nursery, but I dont think the solution is to displace your DD any further. She needs you MORE than ever now. She needs reassurance that she is still loved. She has not been replaced by the baby, you have room in your heart for them both.
You are doing brilliantly with three young kids. Is there anyway you could have somebody come in and mind the baby a little for you now and then so you can spend some quality time with DD? Or, can you try do that when baby is sleeping?
I like the idea of a trip to the park/playground, baby can stay in pram, your dd can play and feel she has special time.
Do you have a baby gym? It wont hurt to let baby lay under the baby gym a little next to you on the floor while you play with DD.
We had the WORST time when I was just at home with them both. Being out helped. Park, shopping centre, a little trip to ELC for the oldest to have a rummage and look at the toys (and sometimes a little treat from the pocket money shelf) and Gymboree. All those things are good for baby too. Grocery shopping even can be made fun if you involve your DD, ask her to help look out for Milk, Apples, etc and let her feal you are talking to her and appreciating your company.
Do you have any friends? Toddler groups? A young baby is very portable, and should also enjoy being taken to a toddler group or coffee mornings.
I hope some of the above helps, but I see you are already given some brilliant advise from previous posters.