What should a working mother of two children do, who has to make up 30 days of fasting, and is unable to fast because she needs to maintain her health for her work with sick children, and because fasting causes her headaches due to her habit of drinking coffee in the morning?

What you mentioned about needing good health during work and feeling headaches due to not drinking coffee is not a legitimate excuse for delaying the making up of missed Ramadan fasts. You must make up what you missed before the next Ramadan arrives to avoid sin and the expiation for delay. You can fast on your days off, drinking coffee at Suhoor to alleviate the headaches. Delaying the make-up fasts without a valid excuse until the next Ramadan arrives necessitates expiation, which is feeding one poor person for each day.