They will last a few hundred cycles and then you're done.
Anything that you rely on in a situation like that should be 100% maintenance free and should not rely on technology more complex than what you can make today in your garage with hand tools. Go over that level and it will just buy you a little bit of comfort but it won't last.
Candles: yes. Solar panels+batteries+inverters: forget it. Even a basic charge controller would be a major feat of engineering under those circumstances. I have some ideas on how I'd make that work if I really had to but I'd probably be too busy defending myself from marauders and taking care of my crops to be able to work on it...
What's interesting here is that plenty of the stuff that we take for granted is absolutely state-of-the-art engineering and there is no way that you'd be able to continue to rely on it for more than the available stock that you could still lay your (likely very grubby) hands on for spares, and even those have a limited shelf life.
Bootstrapping to the present day level of technology where working semiconductors can be taken for granted will take a long, long time. Even if you have the knowledge. Assuming that enough of the eco-system survives that you will be able to survive and procreate I'd estimate at least several generations, if we didn't entirely regress to some more primitive state. Mad Max would be an amazing level to bootstrap from, having working computers means you're focusing on the wrong thing. Electric light might be a good idea, but personally I would not count on LEDs or bulbs to last long enough in quantity and if you're going for batteries keep it simple: lead acid flooded cells (You'll need some basic chemistry tools and materials to make sulphuric acid, but that's probably doable).