iPhone 4 doesn't detect any type of sim card

Hi, I have this phone in a box for about 6 months with no use at all, and my intention was gonna give it to my 3 year old cousin. The problem was when I tried to put the sim card in the phone, this didn't recognize it at all. sometimes it detects it for a few seconds but then the phone turns off and shows the no battery screen. I can't activate it if it doesn't have a sim card. My theory is that the battery (about 4 years old, replaced on a Apple Store) does not have enough voltage or health to power the sim tray and the rest of the phone at the same time. What should I do?