I had to take it with my own SS when he was about the same age as your SD is.  And I had to take it with my wife as well.

Your DW is certainly her mother. However, she is foremost your wife.  Your SD's treatment of your wife is not acceptable. That... is the message they both need.  Regardless of who SD is, she cannot treat your wife disrespectfully. 

Set that mandate, enforce it, and make sure that they both know that your wife will not be disrespected by anyone. Including herself.

DW's naive perspective that she likes that SD needs her is avoidance at a highly developed level IMHO.  DW knows her spawn is using her, taking advantage of her, and your DW does not have the courage to be the adult and hold her kid accountable and end the disrespect.

So, you do it.

 I have.  I had to intervene, give my bride and our kid (my SS-32 who asked me to adopt him 10yrs ago) clarity that no one will disrespect my wife including  my wife and her/our son.

I made it clear to both of them that I would not tolerate it.

Trying to figure out 15 year olds.. esp girls.. haha.. it's a mystery as old as time.. lol.

But.. to your issues here.  At a certain point, you cannot care more than the parent... and your tactic should be more of one approaching your wife (don't try to parent her child.. it won't work as you have found) and not directly trying to parent her daughter.  Correction without connection breeds resentment (and drama) as you have found out.  

It's also important to decide what your hills to die on will be when it comes to her daughter and it's important to understand that she was raised very differently from your own kids.  It's a lot tougher to all of a sudden switch it up on her and have high expectations when she was never faced with the smaller incremental steps over her lifetime.  It can make the changes you want to see seem very drastic.. and it results in her being upset about them (the daughter).. and likely resentful that you are changing her life... in ways she does not see positively.

So, what do you have a say in?  You can have a say when it comes to your own resources.. what your money is spent on. You can have input in how your household is run as a co-head of that household.  But those inputs are things that you and your wife discuss as a couple..it is up to her to implement anything as far as it goes to her daughter.  You cannot and should not be an enforcer of rules for her daughter unless you are in the moment the only person there and you are stepping in to prevent some danger or damage.  

As far as her daughter and what responsibility her mother wants her to have?  That is pretty much up to Mom.  What you can do is encourage her to give her daughter the opportunity to grow and become more independent.  You can encourage her to challenge her child to learn new skills that will benefit her as an adult.. and you can have a say in what the plans are for her daughter's ability to live at home after she turns 18.. ie.. you can set a line in the sand and tell her that you won't allow it... or can negotiate conditions of such residence.. time limits.. obligation to be enrolled in school or other technical training and working.

Try approaching your wife as concerns for her daughter's future happiness in life.  That while you understand that she feels good about spoiling her and being needed.. she won't be there forever and the real gift we can give our kids is the ability to stand on their own two feet.  And.. Rome isn't built in a day.  What are the low hanging fruit here.  What are some things you can frame in positive ways that would make her child's life better.  Like, learning to cook a meal... great skill.. and make her life better in the future.  

As a teen in HS, her primary "job" should be going to school and getting an education to be ready to head off to college or at this point.. start thinking about what other alternatives she might want to pursue (cosmetology.. dental assistant.. nursing).  Participating in extra curricular activities.. sports.. clubs is also enriching and good for kids.  At her age, starting to learn some life skills like how to do a load of laundry.. learning to keep her own space neat... are great too.  Participating in how the household runs is good too... Helping mom prep dinner... clearing the table... taking out the trash.. all small "asks" that mom should be able to make...   15 is a little young to be working probably.. but she may be enticed by the opportunity to earn extra money in another year or so.. again.. working for other people has a way of maturing kids.. who don't have mom and dad there to hold their hands.  But, working is double edged because it often requires the parents to coordinate rides etc.. and if the kid is not naturally great at school.. the time taken can be too much for them.. working can be beneficial.. but there are def arguments on both sides.

In the end.. it's a bit of disengagement..  She isn't your child.. and as long as what she is doing isn't actively harming you or damaging your property.. sometimes we just have to let the parents do what they feel is best.  But, you can encourage positive changes.. if you can frame them so that it comes from a place of caring about her future life and development.

But, I def would back off trying to deal with her daughter directly.. you already have figured out it just causes extra problems.. and it's likely that messages from you are going to be taken with less weight anyway.. and imho having a peaceful homelife would trump fighting with a kid about her "screen time" or whether she made her bed.... neither of which would actually negatively impact my life.

You cannot parent her, and any attempts will be met with much resistance. Her mother is the one you need to work on, as others have suggested. Frame it as you care about SD and want the very best for her, rather than as criticism. 

Her mother is not doing her daughter any favors.

The good news is that as you have success in parenting, your DW cannot ever say "what do you know about parenting anyway?"

OK, where to start. Oh boy. I guess I'm getting flashbacks of my coddled Disney parented SKs SS33 and SD35. Maybe do a search for Disneyland Dad Syndrome. Mums can be like that too. They overly coddle the poopsie so that they'll never leave. The parent is either doing it out of guilt or there was already an issue within the parent that makes them be that way. 

The coddling and doing everything for your SD is only a symptom of a much bigger issue. If you address your DW's doing everything for your SD that she should be doing herself, you'd only be addressing a symptom. The issue lies within your DW. I went through hell living with my DH's deep issue with infantilizing my SKs. If I were to ask you if there are other signs, such as your SD being put above you, your SD coming first, your SD not meeting normal milestones (like getting a teenage job, etc), being clingy with Mummy, what would you say? Overly coddled SKs do not meet milestones at a time that their peers do if ever. Also, don't make the mistake of counting down to 18. As long as your SD is coddled, she's not going anywhere.