Hi @muttling ,
Unfortunately do to the delay in cleaning it after it had been partially submerged in water your console may be damaged beyond economically viable recovery but you never know, you may be lucky. ;-)
As you know water and electronics and electricity are not a good mix.
The impurities in the water causes corrosion and provides circuit paths for the electricity which were not in the console’s operating design and could damage the components.
First do not turn on your console and then you need to remove the battery as soon as possible from the console to minimize any further damage.
Then you need to dis-assemble the rest of the console and clean all the affected parts using Isopropyl Alcohol 99%+ (available from electronics parts stores), to remove all traces of corrosion. Do not use "rubbing alcohol" as in some cases this is only 70% IPA or less, can contain scents and is not as effective. If you have to use it check the label to verify the amount of IPA. The higher the % the better.
Here is a link that describes the process.
Electronics Water Damage
As always with electronics, especially surface mounted PCBs be gentle when handling and especially when brushing away the corrosion. You do not want to remove any components from the board.
Also after this length of time the battery may have to be replaced as well.
Hopefully after you have done all this the console might possibly work correctly again.
Here is a link to the ifixit guide(s) for repairing your console.
Nintendo DSi
If this process seems too daunting, take your console to a reputable, professional electronics repair service and ask for a quote for a repair.