Hello Jen! Try this:- Turn the machine off by holding down the power button turn the machine on and instantly hold down the cmd, alt, p and r. Hold these keys down while it opens up to the apple logo, it will go black and boot again, let it do this 3 times and then when it starts again, let go. See what that does :)

When you say nothing do you mean nothing, no noise or no display? This is very important!

Hey Marco, Yes it should do, It wouldn't be a part of the actually battery as if your actual battery runs out, it wouldn't keep the time anyway. best thing you can do is safely remove the bottom case and it should be fairly obvious thing to spot. Enjoy and good luck!