The rate of my phone battery loss is higher than the charging speed, how will my phone turn on?

I have a Samsung Galaxy Note Edge and its battery has been performing badly recently (i.e. wouldn't turn on unless connected to a charger) but it seems that the rate of the phone battery spending is higher than the rate of my fast charger. I just want to revive it one last time to back up everything to transfer to my new phone. Is there any way I can do from home to be able to do so? Other than buying a new battery?

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