Not going to beat around the bushes here - take it to a repair shop as soon as you can.
The problem with water damage is that it's impossible to diagnose the extent of the damage by just seeing what the average user can do (ie. turning it on and off). It needs to be opened up and inspected by someone who knows what they're doing so they can let you know if it's salvageable, if parts need to be replaced, or if it's a goner.
If the battery icon shows up on screen but doesn't turn on, that tells me that there could be some damage to the board. Best case scenario - it can be saved if the internals are dried out properly (it may need to be taken apart for this). Worst case - the main logic board has some irreparable damage to it from the water and it's either a hefty repair bill and loss of data, or you may want to look into replacement options.