My keyboard has multiple non-pressing keys after cleaning with water and soap

This probably is an issue with water...

Many keyboards have a two plastic layers with conductive traces and pads on them, separated by a third plastic layer with holes under each key (to allow contact). When you then press the key, the two outer layers come into contact with each other, and complete the circuit - a very small movement. See picture below.

It sounds like you used quite a bit of water to 'wash' your keyboard... which isn't really an advisable method. It is highly probable that the water has crept into this three-layer stack (possibly in part due to capillary action).

Unfortunately once the water is in, it is very tricky to get out without separating the layers and gently wiping / drying them. It is quite likely that even leaving the keyboard in the sun for a long time will not get all of the water out.

The presence of water can make it appear that the keys are 'always pressed' (and may then be ignored by the keyboard's controller), and it's also possible that the water could act as a catalyst for corrosion.

I would recommend that you take the keyboard apart, and remove any water (it's not a tricky thing to do... just be careful that you don't tear any of the layers).

I mentioned that the keyboard's controller may ignore keys that are 'always pressed'... Due to the way that the keys are wired, this can have additional (and unexpected) behaviours, such as preventing other keys from working. Look into Key Rollover for more info.