If you can get into the device it should be as simple as de-soldering the two wires connecting the speakers to the main board. Then you would need to get a replacement speaker of the same size and power as the one already in the device which may or may not be a propriety size speaker. The hard part would most likely be figuring out how to open the speaker which I cannot help with since I don’t own one and have never worked on one. This procedure is just how I have changed speakers in multiple different PC speakers or iHome type devices.
I had something similar happen recently to one of my controllers. I did basically the same thing you did only I de-soldered the whole stick to clean the potentiometers, but did not get any improvement. The only thing which worked for me was to de-solder the whole analog stick assembly and solder in a new one. As a side not I also saw a goo on my sticks, but after replacing the stick and getting a better look at the stuff I just decided it was some kind of lubricant from the factory.