If the speaker is simply dirty, clean it and the speaker holes in the case. Then, reassemble it and test the speakers. If they still don't work then continue on.

Once you've removed the motherboard, you are now free to desolder the speaker wires and replace it with a new one.

Make sure to keep the same polarity as the original, as otherwise it may sound a bit different. However, it likely doesn't matter and won't damage anything if the polarity is reversed.