SIM800L LED is not blinking
So I Finally sort out the problem. Both Battery and Module was completely fine. The problem was with the connecting wires. The connecting wires for some reason was not capable to hold the 2amp current. As a solution what I did was, cut the legs of some  quarter watt resistance and solder them to the wire exiting from the battery and fed them directly to the vcc and ground of the module. This way it works perfectly fine.
Even it seems a weird problem with no proper explanation but this solution works for me.
-----------Editing for a better explanation-------------
So my problem was with the connecting wires. For some reason the wires I primarily used could not deliver the the current from battery terminals to the module. May be those wires had a higher resistance for which the current flowing through those could not stay at 2 ampere magnitude and by the time it traveerses the wires and reached to the module, it was no longer at 2 ampere and might be lesser than a certain value which could not trigger the module and so the module did not work.
what I did was cut some legs of a few quatter watt resistance (the metal leads getting out from the carbon resistance from both sides:pic added) and sholder them to the battery terminals. Since I knew the metal legs of quatter watt resistances had a very less resistance from my course that trick worked and delivered 2amp current through them to the module.
I am trully sorry for not attaching picture as I have already submitted the project and currently have no way to replicate the situation.