D.K.
Your friend probably IS a wonderful woman and you are a good friend. But you have to trust your gut. If you feel that you are being manipulated, you probably are. It sure does seem that way to me.
Tell your friend that while you are THRILLED that she has such a talented daughter who clearly has an ear for music, you are just unable to teach at this point.
You are too busy with your family and well, with summer right around the corner - you are SURE that she would understand. Just tell her the truth. You are taking time off from teaching and will be focusing on your own family this year.
Have on hand a list of 5 other people/music stores/learning centers that she can call and wish her daughter the best.
In doing this, you are being true to yourself and being honest with your friend.
You are providing an alternative by supplying her with a list of others that may be able to help out. (Yes - she is on a tight budget and this might not be the ideal option for her - but that is not your problem. You are just trying to be helpful)
You are also excusing yourself from other further obligations while being polite and not harboring bad feelings.
And about the time you put into this child, look at it as an investment in this little girl. Well worth it.
Also, DON'T offer something like "In the fall I might have more time." The mom might take this as a promise and she will keep hounding you. Just go for a clean break while being neighborly.
Then get out of the situation before this gal tries to figure out how else she can take from you!!!!! Enjoy your family and your own quiet time!
Good Luck!
D. in milwaukee