How to get the correct speed/power on downwind?
I spend most of my flying time in C172s of various models as well as 152s, PA-28s and the like. In most C172s you can extend 10&deg; of flap at 100kts, or even 110kts in some models. The white arc is usually for 20&deg; of flap and above and that's 85kts not 80kts, but you should check the POH to be sure or it may be written on the flap handle placard. 10&deg; of flap will certainly help, but it's more your technique that seems to be the issue here. 

Your goal to not add power after you've taken it out is what is the biggest reason for your float, and isn't really constructive. It's normal to reduce throttle to meet a particular airspeed and then throttle up again, if you don't you are relying solely on drag to slow you down, which as you have discovered takes time. Cessnas are draggy beasts because of their low wing loading, wing struts and big rivets, so you can kinda get away with it, but if you tried that in a PA-28 you'd float much, much farther as it is less draggy, so it's not a good habit to get into as you'll be carrying too much extra speed throughout the approach. 

You're never going to hit on the perfect throttle setting that will carry you around the pattern and touchdown at just the right point, there are too many variables to factor in like wind direction, wind speed, rising or falling air, etc. Fixing your throttle forces you to fly a specific pattern for the specific conditions, which won't be the same pattern as everyone else, and it will be different every time. There's no benefit to doing it.   

The moral of the story is don't be afraid to use the throttle as necessary to control your energy, that's what it's for. When you need to reduce speed throttle back more, then increase again when you get to your target airspeed. In the approach make throttle adjustments required to maintain the descent rate you need, if you start to get a bit low add some, if you get to high pull it, and practice practice practice. 

Don't worry about engine failure in the pattern either, if your engine quits you most likely have enough energy to make the field, just clean up and fly your best glide speed. Try it with your instructor: throttle back to idle at downwind and fly a curved approach onto the runway, you'd be surprised how much energy you have at the end.