The charger light goes out when it's done charging. I don't watch mine, but it seems to take just a couple of hours at the most with an old 1.3A wall charger.
There's been a few comments on BF of failed Di2 batteries this summer.
My original battery lasted more than 5 years, and was still working, but I had to recharge it quite often during the final year. The replacement battery has been fine for a few years now. I have about 30,000 miles on the bike now, and everything still works like new.