My kids aren't in their teens yet, but personally speaking I wouldn't give her an allowance. When I went to Uni my parents did give me an allowance as I was living away from home and didn't get anything from the government towards rent / living expenses etc - I also worked small part-time jobs, on and off.
I couldn't have afforded a phone or a gym membership (although did buy a bike!) and must admit that it amazes me that so many students these days can. If I had been living at home (god forbid - couldn't have done that!) I wouldn't have expected anything at all from my parents - and I hated the times I did have to ask for help. But then, I was (and am) fiercely independent and hate having to rely on anyone for anything so am aware I am at the extreme and of the independence spectrum in this respect!
Struggling for money was at times stressful and worrying and I did end up with an overdraft, two student loans and a very hefty 'Career Development Loan' at the end of my 6 years, but it does also give you skills in assessing what is necessary and what is a luxury. The debt never really bothered me; it was structured, sensible and paid off over time. Being in debt, as long as it is structured and has sensible terms, is not the end of the world.
I think it must be very hard to forge an independent life while living at home (very hard to move out of / change the parent / child dynamic) and I think I would try to minimise parental involvement as much as possible. Maybe try something less structured; no regular allowance that she can come to rely on, but the odd 'treat' of a shopping trip for some clothes or books, if you really feel the need?
I just wonder how much she does actually want to be independent if she lets you (effectively) pay for her rent, her household bills, her food, her travel and her phone? Paying only for her own luxuries (gym, phone etc) is not independence. If the amount of the allowance is a 'war' I'm guessing she's not on the side of a lower amount? If she does want to be independent, then she won't accept an allowance; she will walk the walk as well as talk the talk. Independence would be making decisions on what she needs versus what she wants and making a mature decision on whether she wanted to work more hours or get a loan and then managing any resulting debt.
Only you know your daughter and what she is mature enough to handle; hope you find a solution that suits you all.
Blimey that was long - sorry for all the waffle 