As Grumpy_Mike said, the bad news is that while those batteries have a 550mAh rating, you won't get much life out of them the way you want to use them. They are rated at a maximum discharge of 10mA, so you would probably get 55 hours life at that point, but the intended use case is for very low current operation: typically 1mA or less.
I'm already seeing my mistake. Although, if you know of a similar battery (I like them because they are very thin and small), I will appreciate it if you share it.
What you really need to do is go beyond what Arduino environment offers for you by default and dive into microcontrollers power saving modes. Takes bit of learning and experimenting and good meter to measure low currents, but its doable. After that you can start worrying about what sort of battery you need.
If that arduino gadget keeps running all the time, that tiny battery will perform very poorly and you are lucky if it lasts a half a day.
You continually seem to think that the current consumption of a device is expressed in mAH. This is only used to give battery rating not current ratings. For current ratings it is simply mA.
It might not seem a big deal to you but it is showing us that you are very reluctant to relearn things you have understood incorrectly.
This starts to approach quickly a training course of low power design basic's.
For starters you need to measure all current your system eats, sensors, controller, etc...
Then you take a look what you can do to minimize power required by these parts. For example sensor that read once every 10 minutes, probably don't need to be powered rest of 9 minutes and 55 seconds.
Also controllers often have operating modes that minimize power usage.
