Two systems in one key, and only one of them has a battery
Most people treat a key fob as one thing, and that is why a flat battery feels like a broken car. It is two independent systems sharing a plastic shell. The remote is a radio transmitter with a battery, a button pad and an aerial. The immobiliser chip is a tiny transponder with no battery at all — it draws its power from the electromagnetic field of a reader coil around the ignition or the start button. Only the first one can go flat. The second one works for the life of the car, because it has nothing to go flat.
That distinction is worth real money in a car park. If the remote does nothing at all, take the emergency blade out of the shell — press the small release button on the side and the key slides out — unlock the door by hand, then hold the whole fob against the start button and press the brake. On a keyless car that is how the coil reads the chip when the fob has no power to transmit. On a car with an ignition barrel, there is usually an emergency slot or a marked spot on the column. The chip only has to be within a couple of centimetres. The car starts, and you drive home instead of calling a tow truck.
Range is the battery gauge; the buttons are not
The one symptom that names the battery without any doubt is distance. A coin cell does not fail like a switch — its voltage sags slowly, and as it sags the transmitter's output power drops, so the distance at which the car still hears it shrinks. A fob that used to work from a third-floor window and now only works when you are standing at the door is reporting its own voltage. Nothing else on the key can produce that gradient: a switch either makes contact or it does not, and the chip either answers or it does not. Both are binary; the range is analogue. So a shrinking range means the cell, every time.
Buying the replacement is where people go wrong. Read the number printed on the old cell and buy that number, not the one a website lists for your model and year — the same car can ship with different fobs across production runs. The number is a size in millimetres: the first two digits are the diameter, the last two are the thickness in tenths of a millimetre. CR2032 is 20 mm across and 3.2 mm thick. CR2025 is the same 20 mm across and 2.5 mm thick. They look interchangeable and they are not. Fit the thin one where a thick one belongs and the contact is loose — the fob works intermittently, which is far more annoying to diagnose than a dead cell. Fit the thick one where a thin one belongs and the back will not close, or it closes and crushes the clip.
When a new battery changes nothing
Three causes, in order of likelihood. It is not seated properly: the cell is in upside down (the plus face goes to the side the shell marks with a plus), an insulating film is still under it, or one of the three spring contacts is black and not touching. The rolling code is out of step: the remote sends a new number with every press and the car remembers the last one it accepted, so a long spell with a flat battery — or pressing the button dozens of times from too far away — leaves the two counters out of sequence. The car hears a valid signal and refuses it, which looks exactly like a dead fob. The button pad has worn through: the carbon pads under the rubber domes flatten, and it takes a harder and harder press until it takes none.
Re-syncing a rolling code is not one procedure, it is a family of them: ignition on and off then hold the lock button, or a sequence of presses with the door closed, or on a keyless car simply holding the fob against the start button and pressing it. Your own handbook has the exact one on the page about the remote or the battery, and it is worth reading rather than following a generic web answer. When you are stuck, the spare key is the dividing line: if the spare works, the car is fine and the problem is inside this fob (orientation, contacts, sync, pad). If the spare does not work either, the receiver or the immobiliser is at fault, and that is not a kitchen-table job.
Water: the pressing is what does the damage
A fob that has been through the wash is usually not ruined by the water. It is ruined by what happens next, which is people pressing the buttons to see whether it still works. Every press applies power, and power plus water plus two dissimilar metals is electrolytic corrosion — the water alone would dry out, the current turns it into a battery that eats the contacts. So the first move after a soaking is to stop pressing it, then open the shell, take the cell out, blot it and leave it in a dry room for a day or two. Not on a radiator, not with a hairdryer: heat warps the shell and the seal, and warm air drives moisture further into the gaps instead of out of them. Once it is properly dry, clean any green or white fur off the contacts with a cotton bud and a little isopropyl alcohol, fit a new cell, and test. Anything that was soaked in a sweet drink is a different story — sugar water corrodes far faster, and a new shell is often the cheaper answer.