I have an Acer 5516. The battery died quickly and obviously needs to be replaced. However, I was surprised that the laptop does not hold power or work even when it's plugged into the power outlet. Does the computer need a working battery for it to work with the adapter?
Try removing the battery and then see if it'll fire up (it should). How long have you had the laptop? If it has been less than a year, get in touch with Acer, as you should be able to get a new battery under the 1 year factory warranty.