On top of annztoo's excellent advice, I just want to add that spaying plays an important role in reducing cat aggression.

http://www.tenthlifecats.org/all-about-cats/cat-behavior/intra-cat-aggression

Yes, both cats are spayed.

We did the slow introduction when we originally got our kitten
(Zuzu) . Then my daughter was home from school on a one month Christmas break.
After a week they were chasing each other (playfully). Subsequent encounters all
started with hissing then they got to playing after shorter and shorter
intervals (like a few days, then 4, then one day, etc.). I don’t know what went
wrong this time.

They don’t use the same litter box. They have eaten each
others’ food on occasion. Food and water
for each, plus litter box, are kept in closable rooms on opposite corners of
the house.

We have been trading off periods of freedom for both in
turns. My cat is very curious about hers and will go look at her across the
gate, then her cat will often retreat to where she is not visible.

Each cat has matured over time so I'm not surprised at the change in their personalities, especially since they never continued to live together.

Is your cat going up to the gate and just staring at Gigi? If so, that can be another sign of aggression. Plus, the fact Gigi retreats and hides shows she feels threatened and is the submissive one.

Continue to keep them separate and I wouldn't allow Zuzu to even see Gigi at this point since it seems Gigi only wants to hide from her. Illness due to stress could follow if Gigi never has the chance to fully relax and move around the house.

It's great that you're trying to work things out between the two cats!

Her cat has always been shy (of people, too). Plus, this is not the house she grew up being used to. She will have to adjust in the end, though. When my daughter is busy or out of town, the plan was for us to take care of her cat. And vice versa, when we are out.

We brought Zuzu to her apartment once for a short visit, to see if she could be comfortable away from her own house. In that case it was my cat who was intimidated on the other cat's turf and who hid most of the time, even though I and my husband were there.

Gigi is starting to come down and interact with us, but since the spat I make sure my cat is locked up at those times.

How long in general does it take before 2 cats will get along?

According to the ASPCA, eight months to a year.

From their website:

How Long It Takes

According to the ASPCA, it takes most cats eight months to a year to develop a friendship with another cat. Some cats do grow to love each other, but unfortunately, some never even become friends. Some learn to tolerate each other, while others fight every day from their first meeting. The bottom line is that there is no way of knowing exactly how long it will take for your cat to develop a relationship with a new kitten, or how successful it will be.

Im not clear on your set up. I used the "time share" method for about three years when it was clear one of my 2 new cats was always going to be aggressive with my frail elderly female cat (who made it to 22 and is now deceased)

If your set up is such that each cat has her own space or room, with a third space that is a neutral territory that each cat takes turns using. that's ideal and it really is possible to keep this going for years - if it comes to that. It may not. I just put this out there as a worst case scenario and to say that this will make it possible for you not to rush things and to have a more calm household

You will want to have it be a routine - with the switch happening at same times every day. Cats love routine and they'll be happier being confined knowing that they'll have their turn soon enough.

Kasha, that's more or less what I've been doing. Not with a routine, though. Except that her cat gets less free time in the common area. If my daughter is not home then her cat usually wants to stay in her room. If she comes to the wire gate and "asks" to come down, then my cat gets locked up in my bedroom/bath with door closed, as she is able to get over the gate if she feels confined. The other cat has not tried to climb over.

Yesterday they just looked at each other on either sides of the gate for a long time. No hissing.

We bought the claw covers and are going to put them on today.