Project circuit current consumption calculation
I think you are making your calculations more difficult than they need to be.
The average power consumption contributions are:

deepsleep, for 57 out of every 60 seconds: 1.18 mA * 57 / 60 = 1.121 mA
active, for 3 out of every 60 seconds: 55 mA * 3 / 60 = 2.75 mA

average = 3.871 mA
So for over the deeplseep + active period, your battery would last 1100mAH / 3.871 mA = 284 hours.
If you want to calculate the 8 hour operational, 14 hour deepsleep into the calculation you could caculate the contributions as:

8 hour out of 24 hour operation at 3.871 mA = 3.871 * 8 / 24 = 1.29 mA
14 hour deepsleep of 24 hour : 1.121 * 14 / 24 = 0.6539 mA

average = 1.944 mA, battery lasts 1100 mAH / 1.944 mA =  565 hours