Suitable battery for EFCom GPRS/GSM Shield

Hi guys,

I have EFCom GPRS/GSM Shield and Arduino uno to monitor a sensor and send sms if it on or off, and I am planning to power them from rechargeable battery when the wall outlet failure.

What is the best battery type (Ni-MH or Li-ion) and what's capacity needed?

And, is it suitable to connect with **TP4056 charger ** https://www.amazon.com/XCSOURCE-Lithium-Battery-Charger-TE420/dp/B01DRT4PWY/ref=sr_1_1?s=electronics&ie=UTF8&qid=1506001354&sr=1-1 and, MT3608 2A DC-DC booster https://www.amazon.com/UEB-MT3608-Booster-Module-Arduino/dp/B01N3CTAA5/ref=sr_1_1?s=electronics&ie=UTF8&qid=1506001287&sr=1-1

And, how to make the battery operating only when the wall outlet failure?