Hate to say it, but this sounds like typical 9 year old behavior to me! My son is almost 11 and it is just NOW that I can trust him to actually WASH in the shower, clean his nails, brush his teeth, etc.
When he was 9 he would totally do the 'standing in the water' trick too! and fib about being 'clean'. You just have to keep on it- with your SD- I would suggest it is time for a book on hygiene.
I can highly recommend American Girl's 'The Care and Keeping of You' (I work part time at a children's bookstore and this is our go-to book for girls. Be aware though, that it does deal also with having your period and things like that, so you may want to only highlight certain chapters of it right now).
I think sitting down with her and the book for a talk about how as she grows being clean is more important, etc. might help keep her aware of that aspect of things.
As far as cleaning her room, etc. It IS frustrating- but I have to say, we took things away until there was nothing left to take away and it did not seem to help. Using carrots as well as sticks worked much better- not just " I'll take away your phone if you don't clean your room' - but also " You can get that new shirt you wanted at the mall ONLY if you clean your room before we go later today."
Specifics seem to work best- if she does not get the shirt- she knows exactly WHY and what she had to do to get it. It is really like having a toddler all over again, just repeat and repeat. The good news is that in my son's case at least, he seems to have outgrown it mostly by 11!
And- no offense to you AT ALL- but I agree with the poster below that maybe it is time to consider public school for her? It sounds like she has a lot of boundaries to keep straight with you- you are supposed to be her stepmom and her teacher and that doesn't sound like it is working well from a discipline perspective- and also like it is a huge job for you!
LOTS of kids who aren't 'independent workers' LEARN to be so in school. You may still have to put in lots of homework help time- but all parents do, and it will be less than when you were her only teacher.
Also, speaking as someone who has worked with literally HUNDREDS of reluctant readers over the years of all ages and backgrounds- the biggest key to getting a child to read, is getting them to LIKE it. And that can be hard, even when parents are very well-meaning. I would recommend:
1) What are her interests? Take her to the library and ask the librarian to work with your sd one on one to find books SHE will really be interested in. Do NOT criticize or comment on her choices, even if they are not what you would have picked for her. She will be much more invested in reading something she has chosen for herself.
2)Read out loud to her! Honestly, she not too old for it ( I still read out loud with my son!) and most kids love it, even if they won't always admit it. Reading aloud is still the absolute best way to raise a reader- a lifelong reader only comes from a child who learns to love reading for its own sake. Pick something you are both interested in and invest in 20 minutes for a chapter a day. You will be amazed at the change this simple time together can make!!!
3)Do her friends read? See if they can all read the same books. Girls that age want to be included and do what their friends are doing and in this case, peer pressure can be used to your advantage. Harry Potter got SO MANY reluctant readers reading- just because their friends were doing it. But once they got started on it, they were hooked! The key is always to find what really holds her interest.
Good luck - check out public schools near you. I think she needs a little more perspective and challenging from someone besides you!
good luck!