I second that. If you are serious about the quality of the sound, an audio recorder is a must.
You will need to link the video and the audio in an editor but this is relatively easy with your knowledge of the music and the key presses to get that music.
If the only way is through the camcorder, buy a good quality microphone to plug into the camcorder. You will need one with a 12-15 ft lead to get the mic in the piano while the camcorder is aimed at the pianist.Do not use wireless mics.
If you must move round with the camcorder then we are back to the separate audio recorder - or two camcorders, one fixed and one roving. You will need a "constant" with both the video and the audio to align the sound with the action.