The water and alcohol should remove grime that seeps under the key domes. Dirt, grit and etc that is physically stuck will require the wash technique as described in the linked posting.
Try the water and alcohol dropper solution with the batteries out. Shake the water and alcohol out after exercising the key. If you have a hair dryer warm the area around the key. Not to hot but bursts of heat. The idea is not to heat up the calculator but to accelerate the drying.
If you just air dry but are thorough about shaking the water out then a couple of hours should be fine. In fact, if the water drops are confined to the key itself then the shaking or blowing and blotting of the key should suffice for a quick test with the batteries in within minutes of doing the cleaning. Confirm it works then let dry overnight.
If you end up soaking and washing the keyboard under water then the drying period would be a couple of days without a drying box.
Second point about pressure on the key.
Open the back of the calculator. Looking at the calculator inside front at the Circuit Board (CB). Find the area associated with the ON key. Press the key and observe the plastic rivets surrounding the ON key area on the CB. Are these rivets secure or is the CB flexing outward, that is, do the rivets hold the CB in place? If not then it is the rivets which are allowing to much travel for the ON key prior to it contacting the CB. Try pushing on the area of the CB near or at the ON key and simutaneously press the ON key ON, batteries in place. See if this helps in contact. If the rivets flex and the pressing action on the inside of the calculator works then read on.
To fix this, you will have to press down on the Circuit Board and melt the heads of the loose rivets with a flat soldering iron element or hot knife. Heat just enough to soften the rivets, not melt them. Do this while applying pressure.
Another way to enhance the contact between key and board is to place a thin foam cushion in the vicinity of the offending key and the screw the back of the case in place. This to will add pressure and help hold the CB against the keyboard contacts.
Cheers, Geoff