Spilled drink in PS2. What to do?

Because the PS2 is a highly integrated machine, further disassembly will probably do less good than harm.

Depending on how wet you got the thing: To facilitate drying, you can put the unit in a 40°C oven for a while. Think a few hours. A blow dryer might work too, but that is more risky. The point is to get the whole thing warm, not hot, all the way through to help the water evaporate.

If any water got onto the CD-drive assembly, you may have to apply a little bit of silicone grease on the guide rods and spindle. Do not use simple oil, it will harden up and destroy your reader. Rather dont do anything than use the wrong grease.

I would ignore the leftover piece. If it is important, you will notice where it goes.

The steps you've taken, and the steps mentioned by Posipiet for drying, are both good. Rinse the machine with isopropyl alcohol to help cut & remove conductive elements from both the OJ and your tap, and then a final rinse with distilled water. Dry in a 50C oven for several hours and allow to cool to room temperature before powering up.