Hi there, this is Dr Linda.
I'm sorry you've been having this issue and it sounds as though you've been doing a lot of the right things.
It can be hard enough for a cat to get on with another cat but this is especially true when they have encroached on their territory. The key here will be to take things very slowly, be patient, offer reassurance and reward both cats with treats and food.
It can be very tricky indeed to introduce a new cat to a home and for the 'settled' cat to accept them. This is particularly true if the 'settled' cat is used to being 'an only child', and the new cat has no experience of other felines.
There are certainly plenty of things that we can try.
While I understand the desire to introduce them immediately, this process needs to be a very slow one (often taking months).
Separating them at this point is essential to avoid any major fights, and they may need to be separated for some time longer.
First things first, prepare their territories. The new cat should have a designated area that is theirs and theirs alone; an area that has their own food bowl / water bowl / bed / hiding place / toys / litter tray/cat tree etc. We cannot expect the cats to share ANY of their things at this stage, as this will likely only lead to conflict. Similarly, your established cat should have an area that the new cat is not allowed to go in to. When they understand their place in their home, they are less likely to invade each other's territory, creating jealousy.
After a few days, place your new cat in a new area of the house (with all the amenities it needs) and allow the settled cat to enter the new cat's old territory. They will be able to smell them and prepare for their meeting. Wait a few days while your settled cat gets used to the new smell and the idea of a new arrival.
You may also choose to mix their bedding at this stage, so they each sleep on the other cat's old bed.
The next step is to allow them to smell each other through a closed door. If they both tolerate this well, the next day consider allowing them to 'meet' through some form of mesh/cat crate, so they are safe but can 'touch' and smell faces if they like.
We should not hold them, this increases anxiety and does not allow them to run away if they need to.
If at any stage either of the cats show negative behaviours e.g. growling / hissing, take them back to the previous day's exercise. During their 'sniff sessions' it's great if they can each have someone with them who is stroking them and offering them treats so they understand it is all a positive experience.
Once you (and they!) feel ready for a real meeting, give them both access to a neutral room that neither of them sleep in or have belongings in. Ensure they each have an 'escape route', as well as high pieces of furniture or cat trees to run to. Quite commonly at this stage, once cat may choose to run away and hide. This is fine; let them take things at their pace & do not force a meeting.
The first few 'meetings' may only last a minute or two. This is fine, either cat may be finding things a little overwhelming.
Invest in a plug-in adaptor called 'Feliway Friends' which can be plugged in the wall and will release 'Happy cat pheromones' to encourage bonding.
Consider giving your first cat a few weeks of calming supplements in her food to take the edge. These are safe, all natural products that can really help. Examples of brands include Zylkene and Nutracalm, though there are lots out there. These can be given alongside Prozac.
Help both cats feel calm and happy by playing with them regularly (laser pointers, wind up mice, puzzle toys, scenting games etc.) which helps to burn off energy and settle nerves.