Shouldn't Satsuki be a Jimae now? She still has the nemaki/wings at the back but it's already been well over a year (and isn't that Gion Kobu's requirement after a year)?
Someone asked missmyoloko a similar question just yesterday, and a while back, she was also asked about it and speculated about the reasons a bit.
It was said that Satsuki would become jimae after the Miyako Odori, but almost two months have passed and her nemaki still haven’t been cut-off.
It was speculated that the she might not be able to find a suitable place to live, but Satsuki is one of the best earning Geiko in Kyoto, so I doubt that it would take her more than two years to find an appartment.
Then, it was speculated that she wanted to stay in the okiya to stay around Mitsuki, her imouto, to make it easier for both of them to guide her. Mitsuki has already completed her first year as a Maiko in March and has really started to blossom, so I also don’t think that this could be a plausible reason (anymore).
I can only think of one more possibility, and that is that Satsuki might become the atatori, the heir, to the Tsurui Okiya. As an atatori, she would continue to live in the okiya, but she’d also have to be legally adopted by the Tsurui okaasan, and all of the paperwork can take a long time to complete, which would explain the over two years Satsuki has already been a Geiko without becoming jimae. Also, this might have not been clear from the start - the decision might have been made a year after she became a Geiko or a little earlier.
However, this is all just speculation and it has not been proven. I don’t know the real reason and don’t want anyone to state this as a fact or think that it is a fact. There is no explanation for Satsuki not being independent yet, but maybe we’ll get it in the future.