B.S.
I used to be a nanny. One time I got really ill and needed special treatment and was afraid to tell my employer. While I wasn't calling in sick, they did notice my "spark" was gone and asked how they could help. I will always be grateful to them for their understanding that I am a person, not just an employee.
Frankly, because this is so close to your heart, and it's a difficult and emotional situation, so you may be making a lot of assumptions, and you might be too hard on her. It is time to sit down and have a candid talk with her. If you trust her with your child, then you should certainly be able to have a talk with her and come to solutions.
It would be very wrong of you to just let her go without investigating. Don't make any decisions until you do. It's good that you realize the bonding is an issue with making a change. AND, if you make a change you never know....you might end up a far worse situation.
She may be genuinely sick. Maybe something is going on with her and she is afraid to say so, or doesn't know how. Be proactive and ask her how you can help. Maybe she needs a few good days off to rest and recouperate. Knowing you care could help her a LOT. Being a nanny is a tough job and burn out is very high.
Note, if she "calls" you when she is sick, then she's not "not showing up", she is calling you. You might be surprised to learn how many nanny's really do just "stop showing" no call, no message, no information.
Finally, you don't say how "often" she is calling in sick. One time can be a real stress for you. If she gets the flu, it can be many days.....this happens. Are your expectations realistic? Are you overreacting? Is it time to look at some other back-up sources?
Further, maybe you need to create a process for her to call in. For example, if she KNOWS the night before that there will be no way for her to make it the next day (say she has a fever, diarrhea, vomitting within 24 hours of reporting to work time), she could call you the night before and let you know so you have advance planning time. She could also give what I call a "heads up" call (I used this with my preschool staff when I was directing). If there is a possibility she MIGHT be out, she can call the night before and you can create a back up plan in advance instead of being stuck when you are needing to leave for work.
Don't give up on her or your working relationship yet. She is important. Find out what is going on, and together with her make a plan. If she is unhappy and says so, and you both (or you on your own) decide to make a change, you can work together on the transition instead of departing with hard feelings, and your child not getting closure.
COMMUNICATION, COMMUNICATION, COMMUNICATION. It is key, it is essential, it can create success.
Sorry if I sound harsh, but I have a lot of experience with this as a Nanny, and early childhood director, AND a parent. I can see and empathize with all sides. Good luck.