My cat hates my kitten and I don't know what to do

(Note - I don't know if this is relevant but both cats are female)

- One month ago I adopted a kitten (3 months and 2 weeks old).

- My older cat (aged 10) has always been the only cat. She is quite anti-social; she loves my mum, tolerates my sibling and I and despises other cats.

- My family and I tried to introduce them carefully. Aside from an unsavoury encounter on the second day (accidentally left my door open, Kitten ran into Cat, who then attacked her), they didn't interact for the first two weeks or so.

- I followed all the advice online - I kept Kitten exclusively in my room, swapped scents, formally introduced them at opposite ends of the room with Kitten safe in her case and fed them treats afterwards, while they were within sight of each other.

- For a while, I would shut Cat in the study (her fave room she spent 90% of her time in before Kitten) when Kitten was downstairs and Cat wasn't outside, but my mum insisted I stop doing that because it wasn't fair to Cat.

- A few weeks ago, my family got sick of the careful co-ordination act between the two, and insisted that we keep all doors open from then on because they'd have to get used to each other eventually.

- At first, Cat would just stare at Kitten from the other end of the room, often hiss or make threatening meow sounds and Kitten would run off.

- But now Kitten is daring enough to run right up to Cat, even when she's puffed up and hissing.

- Cat is very unhappy. She spends much less time indoors, almost never lets me stroke her (not that she was the keenest on it before, but now she can't stand me) and hisses all the time when she rarely used to before. I feel incredibly guilty because it was my idea to adopt Kitten. I don't regret it per se; I love her deeply and wanted to give her a better home (her previous owner wasn't perfect - they'd feed and shelter her but were unwilling to spend the money to get her vaccinated or spayed) but seeing Cat so mean and angry is awful.

- We need the cats to get along in a matter of weeks or months, not years. My family and I travel a lot, and although we leave the cats in the care of a relative, it'd be a lot easier on all of us if they could peacefully share a house.

- Kitten is playful and adventurous, so confining her to one room again would be cruel. There is no way to separate the upstairs and downstairs levels of the house, plus Kitten's litter box is downstairs so she needs to be able to go there. Once Kitten is spayed we plan to let her go outside, so the cats also must be able to share the garden and the wider outside area.

- Any tips on how to 1. help the cats get along better and 2. improve Cat's happiness?

Thank you!