Background
I have this family member, my cousin, who lives 30 minutes ride away from me. She has a full-time job and her mother takes care of her 6-year-old child during the day, while she is at work. The "taking-care" in this case is mainly taking the child to school. Public transportation to school is not available in my country. Usually, parents drive their kids to school. Recently, for some reason, her mother has to leave town for 2-3 weeks.
This family member asked me to take her child to school for this time period.
Note that I have a full-time job too and a very demanding job as a software developer. I really don't have the time and energy to deal with this situation.
School starts at 8 am and I also have to be at work at 8 am.. So this'd mean for me to maybe arrive 45 min to 1 hour late at work. (Although the child could arrive at school maybe 15 minutes early, this would still make me late for work.)
She probably asked me because she knows that I can substitute my working hours. If I come to work 1 hour late, I can substitute it by working an extra hour in the evening. I have decided not to help her this time because if I say yes, I have to do this for 2-3 weeks, most probably 3 weeks. Coming up to 1 hour late for 3 weeks would not be acceptable by my manager. Plus, I've just finished a project for which I had to work 65 hours a week (for 6 weeks), so I don't have the energy.
There might be other alternatives besides me helping her. My family member has two brothers also living in the area, one of them lives three blocks away from her apartment. So one of them could help, she told me by herself that I'm the first person she thought to ask for help. Also, hiring a person as a nanny, would be an alternative. Finally, she could work something out with her supervisors at work, to allow her to come late at work for these 3 weeks because she has a child to take to school.
Question
How to say NO to this request gently, without hurting her feelings?
I also want to point out that she is divorced and has had lots of difficulties to bear in her life. She is indeed an excellent woman and mother.
Update
I talked with her over the phone and told her: My boss won't be ok with me coming 1 hour later for 3 weeks straight. She said she understood me and will ask another family member, one of her brothers who is married with two kids. He is very much likely to agree to help.
I wanted to be discrete, but now I decided to explain why I was firm to say NO. I am a rational person and I put my work first because I thought that in this situation, she could find a different solution to her problem. I have helped her other times and I am not regretting it but this time I though she asks me for help simply because I don't say "No" to her. And truly, my work situation was that adjusting my schedule would be very difficult for me. Considering everything, I decided that I couldn't agree to my cousin's request.
Thanks a lot for your help!